Answer
$2$
Work Step by Step
$\lim\limits_{x \to 0}\dfrac{e^x-e^{-x}}{x}=\lim\limits_{x \to 0} \dfrac{[(1+x+\dfrac{x^2}{2!}+\dfrac{x^3}{3!}+\dfrac{x^4}{4!}+....)-(1-x+\dfrac{x^2}{2!}-\dfrac{x^3}{3!}+\dfrac{x^4}{4!}-....)]}{x}\\=\lim\limits_{x \to 0} \dfrac{(2x)+2 \times \dfrac{x^3}{3!}+2 \times \dfrac{x^5}{5!}+...}{x}\\=\lim\limits_{x \to 0} (2)+\lim\limits_{x \to 0} \dfrac{2x^3}{3!}+\lim\limits_{x \to 0} \dfrac{2x^4}{4!}+....]\\=2$
