Science:Math Exam Resources/Courses/MATH110/April 2011/Question 02 (a)

MATH110 April 2011

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Question 02 (a)
Evaluate the following limit, if it exists. $\displaystyle \lim _{x\to \pi ^{+}}{\frac {\sin x}{x-\pi }}$

Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Hint
Try evaluating the limit as written. Which rule can you apply to the limit based on the answer you get?

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Solution

Found a typo? Is this solution unclear? Let us know here. Please rate my easiness! It's quick and helps everyone guide their studies. Simply trying to evaluate the limit as written gives the following: $\lim _{x\to \pi ^{+}}{\frac {\sin x}{x-\pi }}={\frac {0}{0}}$ This is an indeterminate form and means that we can use L'Hopital's Rule. Differentiating the numerator and denominator of the function, we get the new limit: $\lim _{x\to \pi ^{+}}{\frac {\cos x}{1}}=\lim _{x\to \pi ^{+}}\cos x$ Simply evaluating this limit gives $\lim _{x\to \pi ^{+}}\cos x=-1$ which is also the value of the original limit.

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