Question

$\displaystyle \lim_{x \to 0}$ $\left(cos\,x+sin\,x\right)^{\frac{1}{x}}$ equals

• A. $e^1$
• B. $e^{-1}$
• C. $1$
• D. $0$

Answer 1

Answer: (A)

$\displaystyle \lim_{x \to 0}$ $\left(cos\,x+sin\,x\right)^{\frac{1}{x}}\quad(1^{\infty}$ form$)$
$=e^{\displaystyle \lim_{x \to 0}\frac{\left(cos\,x+sin\,x-1\right)}{x}}$
$=e^{\displaystyle \lim_{x \to 0}\left[\frac{sin\,x}{x}-\left(\frac{1-cos\,x}{x}\right)\right]}$
$=e^{1-0}=e^{1}$