Determine the following limits $\lim_{x \to 0}\left(\sin x + \frac{e^x-1}{x}\right)$ and $\lim_{x \to 0}\frac{e^x-1}{\sin x}$

Question

Determine the following limits

1. $\lim_{x \to 0}\left(\sin x + \frac{e^x-1}{x}\right)$

2. $\lim_{x \to 0}\frac{e^x-1}{\sin x}$

Answer 1

we have $$\lim_{x\to 0}\left(\sin(x)+\frac{e^x-1}{x}\right)=0+\lim_{x \to 0}e^x=1$$ further we get $$\lim_{x \to 0}\left(\frac{e^x-1}{\sin(x)}\right)=\lim_{x \to 0}\frac{e^x}{\cos(x)}=1$$

Answer 2

Hint. Recall that $e^x=1+x+o_{x\to 0}(x)$ and $\sin(x)=x+o_{x\to 0}(x)$. These asymptotics are derived from Taylor's theorem.

Answer 3

For example, using the definition of derivative:

$$\lim_{x\to0}\frac{e^x-1}{\sin x}=\lim_{x\to0}\frac{e^x-1}x\cdot\frac x{\sin x}=e'(0)\cdot1=1$$

Can you now do the other one?