lim-x-0-tanx-x-1-x-2-

Question Number 155625 by puissant last updated on 02/Oct/21

\lim_{x \to 0} {(\frac{t a n x}{x})}^{\frac{1}{x^{2}}} = ?

Answered by yeti123 last updated on 03/Oct/21

\lim_{x \to 0} {(\frac{\tan x}{x})}^{\frac{1}{x^{2}}} = \lim_{x \to 0} {(\frac{x + \frac{x^{3}}{3} + \dots}{x})}^{\frac{1}{x^{2}}}

= \lim_{x \to 0} {(1 + \frac{x^{2}}{3})}^{\frac{1}{x^{2}} \times \frac{x^{2}}{3} \times \frac{3}{x^{2}}}

= \lim_{x \to 0} {(1 + \frac{x^{2}}{3})}^{\frac{3}{x^{2}} \times \frac{1}{3}}

= e^{\frac{1}{3}}

Commented by puissant last updated on 03/Oct/21

t h a n k s . .

Answered by cortano last updated on 03/Oct/21

\lim_{x \to 0} {(\frac{\tan x}{x})}^{\frac{1}{x^{2}}} = e^{\lim_{x \to 0} (\frac{\tan x - x}{x^{3}})}

= e^{\frac{1}{3}}

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