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Q. $\displaystyle \lim_{x \to \frac{\pi}{4}}$ $\left(tan\,x\right)^{tan\,2x}$ equals

Limits and Derivatives

Solution:

$\displaystyle \lim_{x \to \frac{\pi}{4}}$ $\left(tan\,x\right)^{tan\,2x}$ ($1^{\infty}$ form)
$=e^{\displaystyle \lim_{x \to \pi/4}{\left(tan\,x-1\right)tan\,2x}}$
$=e^{\displaystyle \lim_{x \to \pi/4}{\left(1-tan\,x\right) \frac{-2\,tan\,x}{\left(1+tan\,x\right)\left(1-tan\,x\right)}}}$
$=e^{\displaystyle \lim_{x \to \pi/4}\frac{-2\,tan\,x}{1+tan\,x}}=e^{-1}$