Calculate the limit as x-> pi (cos(x)+1)/(x-pi) using the special limit lim x->0 sinx/x

Question:

Calculate the limit {eq}\lim_{x \to \pi} \frac{cos(x)+1}{x-\pi} {/eq}

Limit:

We will use L-Hopital's rule to solve the problem where we will differentiate the numerator and the denominator and then plug-in the value of x. Here after plug-in the value of x we will get indeterminate form.

Answer and Explanation:

To solve the problem we will use the L-Hopital's rule:

{eq}\lim_{x\rightarrow \pi}\frac{\cos x+1}{x-\pi} {/eq}

If we plug-in the value of x then we will get the indeterminate form:

Now differentiating the numerator and the denominator

{eq}\lim_{x\rightarrow \pi}\frac{-\sin x}{1} {/eq}

Now let us plug-in the value of x as {eq}\pi {/eq}:

{eq}=0 {/eq}


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Understanding the Properties of Limits

from Math 104: Calculus

Chapter 6 / Lesson 5
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