Q. 58 - Calculus

Question

Calculate each of the limits:

$\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{cot x}{\cos x}$.

Step-by-Step Solution

Answer

The solution for the limit $\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{cot x}{\cos x}$ is, $1$.

Step 1. Given Information.

The limit is,

$\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{cot x}{\cos x}$.

Step 2. Finding the solution.

$\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{cot x}{\cos x}$

Inputting $cot x=\frac{\cos x}{\sin x}$,

$$\begin{gathered} =\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{{\displaystyle \frac{\cos x}{\sin x}}}{\cos x} \\ =\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{\cos x}{\sin x \times \cos x} \\ =\underset{x\to \frac{\mathrm{\pi }}{2}}{\lim }\frac{1}{\sin x} \\ =\frac{1}{\sin {\displaystyle \frac{\mathrm{\pi }}{2}}} \\ =\frac{1}{1} \\ =1 \end{gathered}$$