<s>[INST] Prove that $\lim\limits_{x \to 0} \sinh(x)/x =1$. 
Prove that $ \lim\limits_{x \to 0} \frac{\sinh x}{x} =1.$

I am having some trouble proving this without derivative. Some help would be much appreciate!
 [/INST] Rewrite using known limit for $e$:
$$
\lim_{x \rightarrow 0} \frac{\sinh(x)}{x} = \lim_{x \rightarrow 0} \frac{e^x - e^{-x}}{2x} = \lim_{x \rightarrow 0} \frac{e^x - 1 + 1 - e^{-x}}{2x} = \frac{1}{2} \lim \frac{e^x - 1}{x} - \frac{1}{2} \lim_{x \rightarrow 0} \frac{e^{-x}-1}{x}
$$
$$
= \frac{1}{2} \lim \frac{e^x - 1}{x} + \frac{1}{2} \lim_{x \rightarrow 0} \frac{e^x-1}{x} = \lim_{x \rightarrow 0} \frac{e^x - 1}{x} = 1
$$
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