\(\lim\limits_{x \to 0}\frac{e^x-1}{x}\)
\(=\lim\limits_{x \to 0}\frac{[1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+....]-1}{x}\)
\(=\lim\limits_{x \to 0}\frac{1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...}{x}\)
\(=\lim\limits_{x \to 0}[\frac{1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+...}{x}]\)
= 1.