Q. 41 - Calculus

Question

Calculate each of the limits in Exercises $21-48$. Some of these limits are made easier by L’Hopital’s rule, and some are not.

$\underset{x\to 0}{\lim }\frac{\cos x-1}{\sin x}$.

Step-by-Step Solution

Answer

The exact value of $\underset{x\to 0}{\lim }\frac{\cos x-1}{\sin x}$ is, $0$.

Step 1 . Given information

$\underset{x\to 0}{\lim }\frac{\cos x-1}{\sin x}$.

Step 2 . The limit using L'Hopital's rule is given below:

$\underset{x\to 0}{\lim }\frac{\cos x-1}{\sin x}=\underset{x\to 0}{\lim }\frac{-\sin x}{\cos x}$ [L'Hopital's rule]

                       $$\begin{gathered} =-\frac{0}{1} \\ =0 \end{gathered}$$

The exact value is, $0$.