Question

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

Answer 1

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

x → 0
∴ sin x → 0

Let y=sin x

x → 0
∴ y → 0

$$\begin{gathered} \Rightarrow \underset{y\to 0}{\lim }\left(\frac{e^y-1}{y}\right)\times \underset{x\to 0}{\lim }\left(\frac{\sin x}{x}\right) \\ =1\times 1 \end{gathered}$$