Evaluate (x→0)(e^x+e^(-x)-2)/x^2.

Question

Evaluate : 

\(\lim\limits_{x \to 0} \frac{e^x+e^{-x}-2}{x^2}\)

Answer 1

\(\lim\limits_{x \to 0} \frac{e^x+e^{-x}-2}{x^2}\) 

\(=\lim\limits_{x \to 0} \frac{e^{2x}+1-2e^x}{x^2e^x}\) 

\(=\lim\limits_{x \to 0} (\frac{e^x-1}{x})^2\times e^{-x}\) 

\(=\lim\limits_{x \to 0} (\frac{e^x-1}{x})^2\times \lim\limits_{x \to 0}e^{-x}\) 

= 1 × e⁰ 

^{= 1.}