Q. 67 - Calculus

Question

Calculate each limit in Exercises 35–80. 

$\underset{x\to 0}{lim}\frac{1-\cos 2x}{7x}$

Step-by-Step Solution

Answer

The limit is 0

Step 1. Given information

The given expression is $\underset{x\to 0}{lim}\frac{1-\cos 2x}{7x}$

Step 2. Calculation

The limit is calculated as below, 

$$\begin{gathered} \underset{x\to 0}{lim}\frac{1-\cos 2x}{7x}=\underset{x\to 0}{lim}\left(\frac{1-\cos 2x}{7x}\right)\frac{2x}{7x}\times \frac{7x}{2x} \\ =\frac{2}{7}\underset{x\to 0}{lim}\left(\frac{{\displaystyle \frac{1-\cos 2x}{2x}}}{{\displaystyle \frac{7x}{7x}}}\right) \\ \left[\mathrm{From} \mathrm{definition}, \underset{\theta \to 0}{lim}\left(\frac{1-\cos \theta }{\theta }\right)=0\right] \\ =\frac{2}{7}\left(\frac{0}{1}\right) \\ =\frac{2}{7}\left(0\right) \\ =0 \end{gathered}$$