\(\lim\limits_{x\to 0}\left (\frac {\tan x}x\right)^{\frac 1{x^2}} = 1^\infty\)
⇒ \(e\,^{\lim\limits_{x \to 0}\frac 1x \left(\frac{\tan x}x - 1\right)}\)
⇒ \(e\,^{\lim\limits_{x\to 0}}\left(\frac {\tan x - x}{x^3}\right)\) .....(1)
\(\)\({\lim\limits_{x\to 0}}\left(\frac {\tan x - x}{x^3}\right) \;\;\left(\frac 00\right)\)
\(={\lim\limits_{x\to 0}}\left[\frac {\sec^2x - 1}{3x^2}\right]\)
\(={\lim\limits_{x\to 0}}\left[\frac {\tan^2x }{3x^2}\right]\)
\(=\frac 13 \lim\limits_{x\to 0} \left(\frac{\tan^2x}{x^2}\right)\)
\(=\frac 13 \times 1\)
\( = \frac 13\)
From (1),
\(e\,^{\lim\limits_{x\to 0}}\left(\frac {\tan x - x}{x^3}\right)\)
\(= e^{1/3}\)
