Q. 44 - 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.

$\underset{x\to \pi /2}{\lim }\frac{\sin \left(\cos x\right)}{\cos x}$.

Step-by-Step Solution

Answer

The exact value of the limit $\underset{x\to \pi /2}{\lim }\frac{\sin \left(\cos x\right)}{\cos x}$ is, $1$.

Step 1 . Given information

$\underset{x\to \pi /2}{\lim }\frac{\sin \left(\cos x\right)}{\cos x}$.

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

$\underset{x\to \pi /2}{\lim }\frac{\sin \left(\cos x\right)}{\cos x}=\underset{x\to \frac{\pi }{2}}{\lim }\frac{\sin \left(\cos x\right)}{\cos x}$ [ in the form of $\frac{0}{0}$; Using L'Hopital's rule]

$$\begin{gathered} =\underset{x\to \frac{\pi }{2}}{\lim }\frac{\cos \left(\cos x\right)\times (-\sin x)}{(-\sin x)} \\ =\underset{x\to \frac{\pi }{2}}{\lim }\cos \left(\cos x\right) \\ =\cos \left(\cos \frac{\pi }{2}\right) \\ =\cos \left(0\right) \\ =1 \end{gathered}$$

Therefore, the exact value is, $1.$