So, applying this with this particular function we have that: This of course is only valid for values "a" in the domain of the function. Why? Because one definition of a continuous function is that: Since these functions are continuous, and this particular function is defined at the point, this value must be the limit. Now, we just need to remember the definition of secant from trigonometry, and the fact that cos(0)=1: In this case, if we evaluate the function at the point we're taking the limit (in this case 0), we get: Usually, the only case where you'll encounter a discontinuous function in these types of problems is when the function is not defined, because the denominator becomes zero somewhere, or we take the square root of a negative number. In this question we have the limit:Īll of these functions are continuous functions. When we're told to solve limits, the first thing we must do is to try to evaluate the function at the point we taking the limit. Furthermore, this study points to the need of instructors placing strong emphasis on the interrelationship between limit, continuity and differentiability.A Confusing Question to Me: Solving Limits by Continuity I also recommend that instructors use a variety of functions when showing students examples to help students develop their own examples to make sense of concepts, and challenges potential contradictory concept images. Recommendations for teaching include the use both algebraic and graphical forms of function and emphasizing strengths and weakness of each in the context of the problem being solved. This study concludes with directions for future research and implications for teaching. Strong students had a large example space, were able to reason with properties of function, recognized the necessity of reasoning consistently with the algebraic and graphical forms of the same function, and tended to be "smooth" thinkers. Average students were inconsistent in their approach to solving problems, and were very dependent on graphical representation of a function. Weak students had significant difficulty finding the domain of a function. As recent research describes, this study shows that most calculus students only think of functions as chunky, not smooth, when reflecting on change. None of the weak or average students gave responses at the object level and few if any were identified as process. At times students characterized as strong gave responses at the process or object level. This framework provides a tool for classifying the knowledge development indicated by students' responses. Interview responses were interpreted according to the framework of Action Process Object Schema (APOS) Theory. Responses on the written instruments were coded as right or wrong. Data were gathered through administration of the above instruments and one-on-one interviews. In addition there were questions regarding real-world problems. The interview questions were designed to explore how students thought of infinity, function, limit and continuity. Eight participants were later interviewed for the second stage of the study. These results were used to identify participants as strong, weak or average. Students were asked questions in multiple choice and true/false format regarding function, limit and continuity. The research described was conducted in two stages. Yet very few students seem to understand the nature of continuity. Continuity is a central concept in calculus.
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