Q. 68 - Calculus

Question

Calculate each limit in Exercises 35–80. 

$\underset{x\to 0}{lim}\frac{\sin 3x}{5x}$

Step-by-Step Solution

Answer

The limit is $\frac{3}{5}$

Step 1. Given information

The given expression is $\underset{x\to 0}{lim}\frac{\sin 3x}{5x}$

Step 2. Calculation

The limit is calculated as below, 

$$\begin{gathered} \underset{x\to 0}{lim}\frac{\sin 3x}{5x}=\underset{x\to 0}{lim}\left(\frac{\sin 3x}{5x}\right)\frac{3x}{5x}\times \frac{5x}{3x} \\ =\frac{3}{5}\underset{x\to 0}{lim}\left(\frac{{\displaystyle \frac{\sin 3x}{3x}}}{{\displaystyle \frac{5x}{5x}}}\right) \\ =\frac{3}{5}\left(1\right) \\ =\frac{3}{5} \end{gathered}$$