Q. 70 - Calculus

Question

Calculate each limit in Exercises 35–80. 

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

Step-by-Step Solution

Answer

The limit is 4

Step 1. Given information

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

Step 2. Calculation

The limit is calculated as below, 

$$\begin{gathered} \left[\mathrm{From} \mathrm{definition}, \underset{\theta \to 0}{lim}\frac{\sin \theta }{\theta }=1\right] \\ \underset{x\to 0}{lim}\frac{3\sin x+x}{x}=\underset{x\to 0}{lim}\left(\frac{3\sin x}{x}+\frac{x}{x}\right) \\ =\underset{x\to 0}{lim}\left(\frac{3\sin x}{x}+1\right) \\ =3\underset{x\to 0}{lim}\left(\frac{\sin x}{x}\right)+\underset{x\to 0}{lim}1 \\ =3\left(1\right)+1 \\ =4 \end{gathered}$$