Q. 25 - Calculus

Question

Find the limit by hand.

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

Step-by-Step Solution

Answer

On solving the limit, we get, $0$.

Step 1. Given information.

Consider the given question,

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

Step 2. Solve the given limit.

Rewriting the expression,

$$\begin{gathered} =\underset{x\to 0}{\lim }\left(\frac{1}{\sin x}-\frac{\cos x}{\sin x}\right) \\ =\underset{x\to 0}{\lim }\left(csc x-cot x\right) \\ =\underset{x\to 0}{\lim }\left(csc x\right)-\underset{x\to 0}{\lim }\left(cot x\right) \\ =csc 0-cot 0 \\ =\infty -\infty \\ =0 \end{gathered}$$