Question

Evaluate:$\underset{x\to 0}{\lim }\frac{e^{\sin x}-1}{x}$

• A. $0$
• B. $e$
• C. $1$
• D. Does not exist

Answer 1

The correct option is C

$1$

Explanation for the correct answer:

Simplifying the equation to determinate form and applying the limits:

$$\begin{gathered} \underset{x\to 0}{\Rightarrow \lim }\frac{e^{\sin x}-1}{x} \\ \Rightarrow \underset{x\to 0}{\lim }\frac{e^{\sin x}-1}{\sin x}.\frac{\sin x}{x}[\text{Multiply and divide by sin x]} \end{gathered}$$

Applying the limits

$$\begin{gathered} \Rightarrow 1\times 1\left[\because \underset{x\to 0}{\lim }\frac{\sin \theta }{\theta }=1\mathrm{and}\underset{x\to 0}{\lim }\frac{e^x-1}{x}=1\right] \\ \Rightarrow 1 \end{gathered}$$

Thus, $\underset{x\to 0}{\lim }\frac{e^{\sin x}-1}{x}=1$

Therefore, the correct answer is option (C).