Potentials and limitations of GeoGebra in teaching and learning limits and continuity of functions at selected senior four Rwandan secondary schools

Abstract With the use of a variety of digital tools, services, and applications to enhance the learning environment, technology is now deeply interwoven within the educational process. Technological, pedagogical, and content knowledge (TPACK) model was used as a lens to evaluate GeoGebra’s potential and limitations in the teaching and learning of Limits and Continuity of Functions. Intact classes were assigned as experimental and control groups purposively. The study subjects were 252 students and 78 mathematics teachers. The Limits and Continuity of Functions Achievement Test and the Advanced Mathematics Teachers’ Survey were the two data collection instruments. The findings revealed significant differences in conceptual knowledge (t [df] = -11.46, p $ \gt $> .05) and procedural knowledge (t [df] = -11.027, p $ \gt $> .05) between control and experimental group. The output lacks detailed instructions, drawing out some functions requires other content knowledge which is not covered by packages of GeoGebra like the graph of discontinuous functions is not automatically represented in a correct manner are some limitations heighted in this study. Results show the necessity of including GeoGebra in the teaching of Limits and Continuity of Functions.


PUBLIC INTEREST STATEMENT
In this current dispensation, science and technology have become a major focus of discussion in science education.This is attributed to the many and varied benefits that integration of technology into science promises in the teaching-learning engagement.Therefore, the GeoGebra software, as a tool used in easing the construction of knowledge of Limits and Continuity of Functions, a prerequisite to calculus, which happens to be one of the most challenging topics for mathematics majors at university, is considered a golden opportunity.If students should be adequately prepared in the face of the new era of technology to compete successfully in calculus classes at university, it stands to reason why their use of Geogebra in comprehending limits and continuity of functions is critical.This latent goal of this study is to expose learners and teachers to the Geogebra software, and to elicit their sincere appraisals of the use of the software.

Introduction
Beginning in the early 19th century, technology has been incorporated into teaching and Learning.The first automated loom was created by Joseph-Marie Jacquard in 1801, and it was programmed to weave intricate patterns using a punched card (Randell, 2013).Then, this technology was modified for use in educational settings, and in 1834, Jacquard's loom saw its first educational application (Nordhaus, 2006).Since then, technology has gradually been incorporated into Learning environments, beginning with the invention of the chalkboard in the 19th century and finally leading to the widespread use of computers, the internet, and multimedia in the 20th and 21st centuries (Christensen, 2019).The creation of educational software that enables teachers to design multimedia classes and assessments for their learners' dates to the middle of the 1970s.Computers, the internet, and other digital technology were introduced into schools across Sub-Saharan Africa in the 1990s.Students now have access to a wealth of knowledge and instruments for studying because to technological innovation.However, because many families still found the cost of technology to be prohibitive, these tools were not widely used (Roschelle et al., 2000).
In a Sub-Saharan African country like Rwanda, using technology has been agreed as part of the Rwandan education system's effort to promote equitable and quality Learning and improve teachers' capabilities to use digital devices for pedagogical purposes (Mugiraneza, 2021).The Rwandan government's initiated, "one laptop per child" project in 2009.Furthermore, in 2016 ICT policy was revised for meeting established the envisions of "Smart Classroom," a digital education system that enables access to multiple teaching and Learning methods (Kasozi-Mulindwa & Niyongabo, 2018;Wallet et al., 2015).
Many recent studies (Uwurukundo et al., 2022;F. S. Yerizon et al., 2021;Zulnaidi et al., 2019) have emphasized on the use of the dynamic mathematics software "GeoGebra".This software is free at GeoGebra's official (www.geogebra.org)and are compatible with Windows, Macintosh, Linux, and Unix platforms, Android, iPad, and web.For improving the use of GeoGebra, the International GeoGebra Institute (IGI) was founded at the end of 2007 with the four goals of providing teacher training and support, developing teaching materials and software, conducting research, and contacting underprivileged communities to better assist GeoGebra communities through their rapid growth (Hohenwarter & Lavicza, 2010).
The GeoGebra software, which is the focus of this study, has become a reliable resource for learning calculus.GeoGebra as a Computer Algebra System (CAS) includes algebraic definitions of functions, equations, and coordinates, as well as symbolic representation, whereas GeoGebra as dynamic geometry software (DGS) includes visual representations.To establish hypotheses or draw generalizations while keeping track of the change in properties, DGS supports students in determining the properties of the figure they have created, discovering the shapes, and interacting with the unchanging and changing properties of the figure (Dana-Picard & Kovács, 2021).Using GeoGebra in calculus enhances the Learning and teaching process by helping teachers design effective instructional lessons.For students interested in mathematics, engineering, finance, physics and biology, calculus, and analysis are integral parts of these courses (Baleanu et al., 2019).
According to several studies, students struggled to learn calculus (Arini & Dewi, 2019).The challenges faced by these students were overcome in several ways.Case and Speer (Arini & Dewi, 2019) investigated the methods that teachers could employ to help calculus students comprehend abstract concepts and formulate arguments using calculus theorems.Adams and Dove (2018) implemented flipped learning to enhance students' Learning outcomes and perspectives on learning calculus.A 2014, study conducted by Dawkins and Epperson examined the problem-solving-focused laboratory (PSFL) used in traditional calculus instruction.The material in calculus remained abstract; hence, the outcomes have not been great ((S.F. Yerizon et al., 2021).An alternative approach to reduce those challenges of abstractness of calculus is provided by using GeoGebra (Tatar & Zengin, 2016).But students still have difficulties on the using GeoGebra to learn calculus.
Several strategies can assist students in using software to learn calculus.First, it is crucial to include online discussion boards, text-based tutorials, and videos.Additionally, giving students practice problems and interactive exercises specially designed to teach them how to use the program for becoming more familiar with it and improve their understanding of calculus principles.Finally, having an instructor or tutor on hand to respond to queries and offer specialized counsel will help students overcome any difficulties that they may experience (Pereira et al., 2022).
How technology is used and how it can be implemented in a particular area of learning content are both factors that determine how effectively it is used.It is assumed that teachers who possess TPACK and proximities experiences can choose which pedagogical approach to employ and how to incorporate technology into a specific subject area (Muhtadi et al., 2017).This study highlighted the potentials and limitations of using GeoGebra in teaching Limits and Continuity of functions of one variable as one of the key concepts in calculus, also this study investigated the impact of TPACK in reducing the limitations of using GeoGebra on students' achievements in Learning Limits and Continuity and examining GeoGebra's impact on students' procedural and conceptual knowledge of Learning Limits and Continuity.Besides that, we cannot ignore the role social proximity as one of the factors influencing the effective use of GeoGebra toward achievements in Learning Limits and Continuity of functions.
Many mathematics teachers face difficulties demonstrating calculus on a blackboard because of its dynamic nature.In addition, students were confused about the image of a function at a point and its limit confusion about piecewise functions by solving inequalities, and they could not find the limit of a function from a graph.They also have difficulty sketching trigonometric, piecewise, exponential, logarithmic, and identifying the domain of functions as fundamental calculus concepts (Thompson & Harel, 2021).Hohenwarter et al. (2008) clarified the package of this software, which can help to reduce some challenges in derivatives, limit of function at points, domain of function through a dynamic approach, sketching a function given in polar or parametric functions, trigonometric functions, and other difficult functions.In addition to the GeoGebra package being unable to provide immediate feedback to students when the class size is reasonable, students can also validate their own work (Wassie & Zergaw, 2019).GeoGebra simplifies the evaluation process by creating instructive practice sheets or applets that include provoking questions.Mathematics education now includes geometric and algebraic study tools.
With GeoGebra, students' greater access to software offers engaging and effective Learning settings that encourage student engagement and success (F. S. Yerizon et al., 2021).For instance, it is easier to understand that the graph represents a continuous function through the DGS.Through visualization, students were encouraged to participate in the class and aid in the clarification of the theorem.This helps students develop their creative thinking skills by discovering mathematical truths and relationships (Uwurukundo et al., 2022).Along with the concept of Limits, derivatives, and Continuity of functions, the DGS provides dynamic relationships between these concepts.CAS and DGS both use GeoGebra to make a relationship between geometry and algebra (Körtesi et al., 2022;Zengin et al., 2012 ).
Students are motivated to engage in the lesson when using this approach to clarify the theorem because of the multiple representations supported in GeoGebra, including geometric, algebraic, and calculus commands (Zulnaidi et al., 2019).

Study objectives
The study was specifically guided by the following objectives.
(1) To examine the effectiveness of using GeoGebra for enhancing students' achievements in conceptual and procedural understanding of Limits and Continuity of Functions.
(2) To examine the impact of Technological, pedagogical, and content knowledge (TPACK) on reducing the limitations of using GeoGebra for students' achievements in learning Limits and Continuity of Functions.

Theoretical framework
This study uses the framework developed by Mishra andKoehler's (2006) improved Shulman's (1986) characterization of the teacher knowledge framework, which states that teaching and Learning should integrate technology.These authors established TPACK by explicitly considering the role of technical knowledge in effective teaching (Koehler et al., 2014).The TPACK framework focuses on the relationships between constraints on and opportunities provided by the three bodies of knowledge.
The TPACK framework comprises three main parts (TPK), content knowledge (CK), and technological content knowledge (TCK).For successful technological integration, each of these interrelated and mutually beneficial components is required.The term technology-enhanced teaching and Learning (TPK) refers to the understanding of teaching and learning with technology, including the capacity to evaluate, choose, and employ the best technological tools for the task at hand.CK refers to an understanding of the topics being taught, including language arts, arithmetic, science, and social studies.The ability to effectively represent, access, and manipulate content through technology is termed TCK (Bwalya & Rutegwa, 2023).Shulman created PCK in 1986, which is a comparable but more extensive structure.TPK and TCK are both present in PCK, but this approach places more emphasis on the contextual factors that affect instruction, such as students, curriculum, setting, and culture.TPACK is more concerned with the content and tools used to teach, whereas PCK is more concerned with teaching and Learning (Rakes et al., 2022).
To effectively prepare and organize lessons, teachers must be knowledgeable about pedagogy, content, and technology (Elas et al., 2019).This means that teachers should understand the concept to be taught to learners, when to teach it, and the steps involved in delivering it.Technology acts as a supporter to visualize content, and this happens when teachers understand the usage of technology tools.Therefore, for successful use of GeoGebra, teachers should be empowered by the TPACK framework (See Figure 1).

Potentials of GeoGebra in the teaching-learning limits and continuity
Many mathematics teachers face difficulties demonstrating calculus on a blackboard because of its dynamic nature.In addition, students were confused about the image of a function at a point and its limit confusion about piecewise functions by solving inequalities, and they could not find the limit of a function from a graph.They also have difficulty sketching trigonometric, piecewise, exponential, logarithmic, and identifying the domain of functions as fundamental calculus concepts (Thompson & Harel, 2021).Hohenwarter et al. (2008) clarified the package of this software, which can help to reduce some challenges in derivatives, limit of function at points, domain of function through a dynamic approach, sketching a function given in polar or parametric functions, trigonometric functions, and other difficult functions.In addition to the GeoGebra package being unable to provide immediate feedback to students when the class size is reasonable, students can also validate their own work (Wassie & Zergaw, 2019).GeoGebra simplifies the evaluation process by creating instructive practice sheets or applets that include provoking questions.Mathematics education now includes geometric and algebraic study tools.
With GeoGebra, students' greater access to software offers engaging and effective Learning settings that encourage student engagement and success (F. S. Yerizon et al., 2021).For instance, it is easier to understand that the graph represents a continuous function through the DGS.Through visualization, students were encouraged to participate in the class and aid in the clarification of the theorem.This helps students develop their creative thinking skills by discovering mathematical truths and relationships (Uwurukundo et al., 2022).Along with the concept of Limits, derivatives, and Continuity of functions, the DGS provides dynamic relationships between these concepts.CAS and DGS both use GeoGebra to make a relationship between geometry and algebra (Körtesi et al., 2022;Zengin et al.,  Students are motivated to engage in the lesson when using this approach to clarify the theorem because of the multiple representations supported in GeoGebra, including geometric, algebraic, and calculus commands (Zulnaidi et al., 2019).Through the graph, it is easier for students to conclude that it represents a discontinuous function because there is a jump between the two functions.In addition, using the GeoGebra package related to the Limits and Continuity concepts, it is simpler for learners to see that the definition of a continuous function is not verified.

GeoGebra packages allow learners to show the relationship between domain, vertical asymptote and continuity at point or interval
, which means that it is undefined because on the left of two there is a point of discontinuity that is one or −1.In addition, at point x = 1 or −1 the graph exhibits  a vertical asymptote.Therefore, the function is not continuous at an interval À 2; 2 � ½ when the domain is restricted.

GeoGebra packages allow learners to compare the differentiability and continuity theory of function
One of the most crucial topics is Continuity and differentiability, which aids students in understanding ideas such as Continuity at a point, Continuity on an interval, and derivative of functions.However, maintaining the Continuity and differentiability of functional parameters (Ciesielski & Seoane-Sepúlveda, 2019).Differentiability implies Continuity but Continuity does not imply differentiability.A continuous function is defined for all values in its domain and is not discontinuous at any point in its domain.A differentiable function has a derivative at each point in its domain.
A continuous function can be differentiated.Many Learners have misconceptions in which they mention that this theory is the opposite.GeoGebra helps clarify the relationship between the differentiability and Continuity of the function.Besides the potentials of using GeoGebra in teaching some concept of calculus like Limits and continuity of functions.There are other potentials in Geometry and mathematical modelling.

Using GeoGebra to sketch some surface in 3D
This figure 7 represent how students can be motivated when drawing the torus with r1=−1 and r2 = 3, . Sketch torus using GeoGebra.
When using the animation of GeoGebra all parameters are varying and torus may gives different surface, see on figure 8 below.Through that animation can make students to be attentive and motivated.The surface below is derived from the torus with u ¼ 0:71; v ¼ 0:005; r1 ¼ 0:48;

Modeling and programming in GeoGebra
In accordance with Dundar et al. ( 2012), The conversion process is a constant and crucial part of realistic mathematical settings, and mathematical modeling includes it.Modeling is the process of transforming mathematically based accidents and incidents that actually happen into a real-world setting.GeoGebra programming can be used to create a string with a parabola shape by utilizing the vector applications and sequence concept.This can inspire student to discover how mathematics can be enjoyable and help make real-world language more real.
Figure 9 below shows how packges of GeoGebra can be used to modele parabola into string.

Limitation of using GeoGebra in teaching limits and continuity
We appreciate the tremendous contributions that professionals have made to the development and design of GeoGebra.However, while considering the significance of GeoGebra usage in teaching and Learning Limits and Continuity of functions, we encountered several restrictions with GeoGebra 6.0.It is not always easy to use commands in the input bar, particularly for people without a programming background.The output lacks detailed instructions, missing functions such as asymptotes, inclusion, or exclusion of the boundary for piecewise functions, and drawing out some functions requires abilities not covered by GeoGebra.

GeoGebra packages provide the solutions that are missing steps
Helped by CAS commands, values for Limits and derivative of functions may be obtained easily which affect algebraic manipulations, rules, and properties.Proving final answer without detailed may affect the students to memorize the properties or the rules of the concepts (Ruthven, 2002).
Figure 10 represents the missing steps for finding the limit of the function and missing steps in finding the derivative.Missing steps can provoke students to forget rules or some algebraic manipulation.

GeoGebra packages confuse the learners on some graph of removable discontinuity
Helped by the DGS, some functions that represent removable discontinuity behave like continuous functions.This can create the confusion on students because some graph in GeoGebra represents discontinuous function as continuous function.
Figure 11 shows how GeoGebra packages can confuse learners if they have insufficient content knowledge on Continuity of functions.

Figure 10. Missing steps in finding limits and derivatives of functions using GeoGebra.
The graph in green shows that GeoGebra lacks the ability to automatically represent some graphs of discontinuous functions.It is sometimes misleading to visually characterize discontinuities from graphs unless the scale is adjusted.When sketching these functions using GeoGebra, f x ð Þ ¼ x 2 À 1 xþ1 and g x ð Þ ¼ x À 1 you get the same graph is obtained, but this is not true.Therefore, teachers should have CK before using technology, for function f x ð Þ ¼ x 2 À 1 xþ1 has points −1(see the figure in blue), which does not define the function, while function g x ð Þ ¼ x À 1 is defined at all points (see the figure in green).GeoGebra was simplified prior to the sketching.
which is not the case.Teachers with CK of rational functions can help students to mention that at x ¼ À 1; f x ð Þ will have points of discontinuity.

GeoGebra packages allow confuse learners on some graph of non-removable discontinuity
Some piecewise functions seem continuous through visualization using GeoGebra but are not continuous using the definition of continuous functions.It is important for teachers to have CK before using GeoGebra.Figure 12 shows how GeoGebra packages can confuse learners if they have insufficient content knowledge on sketching graph and its Continuity at point.The graph in red shows that the graph represents a continuous function, but it is not correct because, based on algebraic computation, the definition of a continuous function is not verified.lim The blue graph indicates that the function is discontinuous at x = 0. To demonstrate that the blue graph teacher should be equipped with the TPACK.

GeoGebra package missing information on inclusion or exclusion of inequality
For the piecewise function, teachers should have CK related to the interpretation of the interval, that is, exclusion or inclusion of points.GeoGebra does not exclude open points as closed or open points.
Given Missing n left or extra n right 2x þ 1; x � 5 x þ 2; x<5 � , sketching this function using GeoGebra, there is no difference for strictly greater or greater or equal for teacher who are not aware about the CK on piecewise function.Some piecewise functions are represented without considering the inclusion or exclusion of the boundary.Teachers equipped with TPACK will help students to see if x ¼ 5andf x ð Þ ¼ 11and point 11 will be included, while x ¼ 5andf x ð Þ ¼ 7, point 7 will be excluded.
Figure 14 shows how to graph piecewise function using GeoGebra when you are equipped with content knowledge.

Study design
A quasi-experimental design with non-equivalent groups was used in this research.This means that before the experiment, students in both experimental and control groups were subjected to pretest to check their previous background in the subject.Following the pretest was a six-week period of intervention, with three hours contact periods per week.The experimental group was taught using the GeoGebra software, while the control group received instructions using traditional techniques.All tests were simultaneously administered to both study groups.

Study populations and sample
This study comprised 252 students between 16 and 18 years of age from nine secondary schools.Students were categorized into control and experimental groups (see Table 1).In addition, this study used 78 teachers who have experience of more than three years and were trained and aware of using GeoGebra in daily teaching activities (see Table 2).
Intact classes were assigned as experimental and control groups purposefully.Students who studied a combination of computers and mathematics were placed in the experimental group, while those who studied mathematics devoid of computer science in combination constituted the control group.

Research Instruments
Two instruments used in the study were the Limits and Continuity of Functions Achievement Test (LCFAT) and the Advanced Mathematics Teachers' Survey (AMTS).This achievement test was developed based on questions in students' mathematics books prescribed by the Rwandan Education Board.The teachers' survey was adapted from a short assessment instrument for TPACK (Schmid et al., 2020).
Both instruments were pilot-tested just after validation by peers and experts in the Testing and Evaluation Department of the University of Rwanda's College of Education to ensure their reliability.The Cronbach's alpha reliability coefficients for the LCFAT and AMTS were computed to be 0.7586 and 0.724 respectively.An experimental group was provided with instructions using a single GeoGebra file which contained sketching and study the Continuity of functions (Polynomial, trigonometric, rational, irrational, and piecewise), the image and Limits of the point at a function.The graphical features of GeoGebra helped the students gain conceptual and practical knowledge of Limits and Continuity of functions.

Intervention
Students in the experimental group cooperated in groups to complete in-class exercises.In addition, they were given assignments that piqued their desire to learn more about the Limits and Continuity discussed in class using mathematical software.The teachers' were knowledgeable in the use of the GeoGebra software for teaching their lesson.The students in the control group received traditional teaching strategies.A step-by-step procedure for solving problems on Limits and Continuity was provided.The experimental group used the GeoGebra Software to complete the Limits and Continuity tasks.

Data analysis
The achievement test data were analysed using both descriptive and inferential statistics in SPSS version 25.0.Independent samples t-test was used to determine whether there was a statistically significant difference between the experimental and control groups before and after intervention.All computations of t-test analysis were done at α = .level of significance.Data from the survey questionnaire were analyzed using descriptive statistics.

Effectiveness of using GeoGebra for enhancing students' achievements in learning limits and continuity of functions
The pretest mean scores of the control and experimental groups were compared using an independent t-test.
According in Table 3, a mean score of 11.82% was achieved in the experimental group compared to 11.47% in the control group.There was a 0.0035 difference in the mean scores across the groups, with a t value of 0.447 (t< 1:96 j jÞ.However, the p-value was 0.613 (p > .05),implying that the mean scores of the two groups did not differ significantly.This finding suggests that the skills of students in the control and experimental groups were comparable before the intervention.
According to Table 4, a 55.82% average was achieved by the experimental group, whereas only a 23.22 % average was achieved by the control group.The average difference in scores between the groups was 32.60%, with a t -value of 19.922 ( t j j>1:96Þ.There was a low p-value (p<.05), indicating that the two groups' means were considerably different from one another.By using GeoGebra, the experimental group outperformed the control group of students in terms of academic performance, as shown by this result.The experimental group fared better on the post-test than the control group.
A paired sample t-test was used to compare the outcomes of the pretest and posttest for the experimental and control groups.The results, as given in Table 5, show that the experimental group's mean score difference between the pretest and posttest was 44.003%, compared with 11.757% for the control group.The experimental group's t value was 29.705 and the p-value was 0.00, both of which were low (p < .05),demonstrating that the scores from the pretest and posttest were significantly different.There were significant differences between the pre-and posttest scores for the control group, as indicated by the control group's low (p < .05)t value of 19.818, and p-value of 0.00.This shows that the test results for both the experimental and control groups increased significantly.These results demonstrate that students benefited from both approaches, although the experimental group's mean difference or improvement in scores was greater than that of the control groups.

Comparison of effects of using GeoGebra in enhancing students' procedural and conceptual knowledge in experimental and control groups
To assess the conceptual and procedural understanding between the two groups, an independent t-test was performed.The results revealed no noticeable difference in conceptual and procedural knowledge between the two groups before the intervention.(Conceptual knowledge, t = −0.346,i.e., t j j<1:96, p-value is 0.242, i.e., p > 0.05; see Table 6 , (procedural knowledge, t = 0.664, i.e., t j j<1:96, p-value is 0.365, i.e., p > 0.05, see Table 7).If the groups are equal at the outset, the experiment may be conducted, and the conceptual and procedural understanding of mathematics on the Limits and Continuity of function of one real variable can then be compared.
Student mathematical conceptual understanding was significantly different between groups, according to an independent t-test analysis (t = −11.460,p < 0.05; see Table 8)).This shows that the experimental group had greater conceptual knowledge than the control group.There was a substantial difference between the groups in terms of procedural understanding of mathematics, according to an independent t-test study (t = −11.027,p < 0.05).Observe Table 9).This shows that compared to the experimental group, pupils in the control group had less procedural knowledge.

The impact of TPACK on reducing the limitations of using GeoGebra on student's achievements in learning limits and continuity of functions
Based on the responses in Table 10, most agreed that Technological and Pedagogical Content Knowledge influences the selection of technology that can enhance students' outcomes.

Discussion of the results
The experimental and control groups' outcomes were significantly different, according to quantitative statistical analysis (Table 3).In addition, the findings revealed a significant difference in the conceptual and procedural knowledge of the Limits and Continuity of the functions (Table 8 and (b).There were significant differences because students in the treatment group had the opportunity to use both traditional approach and the GeoGebra Software.As the GeoGebra package enabled students to receive feedback immediately, they compared their results obtained from GeoGebra software and the traditional approach.They completed more activities, which helped improve their academic achievements.Students in the control group had greater procedural understanding before the intervention.According to a comparison of the results from using GeoGebra and handwriting, students in the experimental group had higher procedural knowledge than those in the control group after the intervention.This finding may have been influenced by the social proximity of students, as discussed in the groups.
Students in the experimental group had more conceptual and procedural knowledge in the posttest than those in the control group.A paired sample t-test (see Table 5) was used to determine the statistical significance of the mean difference between the pretest and posttest scores for both groups.The results showed that each student's posttest score was significantly higher than their pretest score for both groups.Each method improved the performance of the students, but the integration of GeoGebra improved more in the experimental group than in the control group.
By enabling users to see, investigate, and understand the principles of Continuity, GeoGebra support in the improvement of procedural knowledge.Since the software offers a variety of tools and features, users can investigate and interplay the relationships between algebra, geometry, and calculus.When students plotting function can use the features of GeoGebra to test its Continuity by computing of Limits and image of function at given points.In this case, student can compare the plotted graph and computed values to check the Continuity of function at given point or interval.Using GeoGebra learners explored the notions of the Continuity concept in a way that is more engaging and intuitive than traditional teaching techniques.This is influenced by feedback from computed results and real-time graphing.When students notice that the answer is incorrect, they investigated where the error is, by comparing the graph and using algebraic computation.
Additionally, being able to alter equations and objects in real-time helps to reinforce the procedural and conceptual understanding required to solve mathematical problems.This result also is supported by the findings of Zulnaidi and Zmri (2017) on the effectiveness of GeoGebra software "the intermediary role of procedural knowledge on students' conceptual knowledge and their achievement in mathematics".And the results of the study conducted by Ocal 2017 on the effect of GeoGebra on students' conceptual and procedural knowledge of the application of derivatives.Unfortunately, the result from Zulnaidi and Zmri (2017) and Ocal (2017) did not considered the impact of TPACK and the factor of proximities toward those achievements, procedural and conceptual understanding.
The results in Table 10, show that most respondents concurred that teachers' knowledge of pedagogy, subject matter, and technology is necessary for GeoGebra to be successfully used in educational settings.If the teacher is familiar with TPACK and its underlying theories, GeoGebra might help students to better understand the Limits and Continuity of Functions concept.Teachers who are equipped with TPACK and social proximities help students to be more attentive in class because of their curiosity about using GeoGebra to visualize graphs of continuous and discontinuous functions.GeoGebra not only makes learning easy but also keeps learners interested and motivated when teachers are aware of the content and pedagogy related to the topics.

Conclusion
This study attempted to determine the effectiveness of using GeoGebra for enhancing students' achievements in Learning Limits and Continuity of functions, compare the effects of the use and non-use of GeoGebra in enhancing students' procedural and conceptual knowledge in limits and continuity of functions, and the impact of TPACK on reducing the limitations of using GeoGebra for students learning achievements in Limits and Continuity of functions.In conclusion, this study has established the effectiveness of the use of Geogebra for learning achievement in limits and continuity of functions.However, it has also been found that the traditional use of paper and pen method also leads to a learning achievement, although the use of Geogebra software leads to greater learning achievement in both procedural and conceptual understanding of the concept.Also, the advanced mathematics teachers in secondary schools believe very strongly in the use of TPACK to greatly reduce the limitations in conceptualizing limits and continuity of functions.

Implications for further study and recommendations
The outcome of this study implies that the use of Geogebra is an important way to achieve both procedural, conceptual and overall understanding of limits and continuity of functions.Where possible, teachers who are trained in the use of the software should be encouraged to use it.However, the outcome of this study also implies that in the absence of the software teachers can still effectively use the traditional paper and pen method for achieving some reasonable level of learning achievement in their limit and continuity classes.For further investigations, it is recommended that the use of GeoGebra for improving the proximities in educational sciences especially mathematics be explored.We suggest the improvements to the GeoGebra package to make it easier to use in a variety of mathematical situations.In addition, we encourage teachers and learners to use GeoGebra apps on smartphones and tablets in areas where it is difficult or impossible to access a computer or the internet.Lastly, teachers should apply GeoGebra as a TPACK tool to help students understand complex problems, but not use GeoGebra to replace conventional method.

Figure 2
Figure 2 below shows how you may use GeoGebra Packages to find limit of function.

Figure 3
Figure 3 below shows how you may use GeoGebra Packages to verify the continuity of function.

Figure 4
Figure 4 below shows how you may use GeoGebra Package to verify the discontinuity of function.

Figure 5 Figure 3 .
Figure 5 below shows how you may use GeoGebra Packages to verify the continuity of function on the interterval.Example: Discuss the Continuity of f(x)= 1x 2 À 1 over the interval À 2; 2 � ½

Figure 4 .
Figure 4. Verify the discontinuity of function. lim

Figure 5 .
Figure 5.Using GeoGebra to show the relationship of domain, asymptote, and continuity of function.

Figure 6 .
Figure 6.Using GeoGebra to verify the relationship between differentiability and continuity of function.

Figure 6
Figure 6 below shows how you may use GeoGebra Packages to compare the continuity and differentiability of function at point.

Figure
Figure 8. Animation of torus with variation of parameters.

Figure
Figure 11.Misrepresentation of function discontinuous function using GeoGebra.

Figure
Figure 12.Misrepresentation of discontinuity of piecewise functions using GeoGebra.

Figure
Figure 13.Misrepresentation of graphing piecewise function using GeoGebra.

Figure
Figure 14.Graphing Piecewise function using GeoGebra with equipped with TPACK.

Table 2 . Characteristics of teachers Gender Number of teachers Percentage
Figure13shows how GeoGebra packages can confuse learners if they have insufficient content knowledge on sketching graph of piecewise function.