Q. 20 - Calculus

Question

Limits of basic functions: Fill in the blanks to complete the limit rules that follow. You may assume that k is positive. 

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

Step-by-Step Solution

Answer

The value of the limit is $\underset{x\to 0}{\lim }\frac{1-\cos x}{x}=0.$

Step 1. Given Information.

The given limit function is $\underset{x\to 0}{\lim }\frac{1-\cos x}{x}.$

Step 2. Filling the blank.

To fill in the blank, apply L'Hopital's Rule.

So,

$$\begin{gathered} \underset{x\to 0}{\lim }\frac{1-\cos x}{x} \\ =\underset{x\to 0}{\lim }\frac{\sin x}{1} \end{gathered}$$

Put the value $x=0$

$$\begin{gathered} =\frac{\sin \left(0\right)}{1} \\ =0 \end{gathered}$$