1-lim-x-e-x-e-x-2-x-2-lim-x-pi-2-cos-x-x-pi-2-3-lim-x-0-1-tan-x-x-2-1-sin-x-x-2-

Question Number 112273 by bemath last updated on 07/Sep/20

(1) \lim_{x \to \infty} {(e^{x} + e^{- x})}^{\frac{2}{x}} = ?

(2) \lim_{x \to \frac{π}{2}} {(\cos x)}^{- x + \frac{π}{2}} ?

(3) \lim_{x \to 0} \sqrt{\frac{1 + \tan x}{x^{2}}} - \sqrt{\frac{1 - \sin x}{x^{2}}} ?

Commented by Dwaipayan Shikari last updated on 07/Sep/20

e^{- 2} {(1 + e^{2 x})}^{\frac{2}{x}} = y

- 2 + \frac{2}{x} l o g (1 + e^{2 x}) = l o g y

- 2 + \frac{4 e^{2 x}}{1 + e^{2 x}} = l o g y

- 2 + 4 (1 - \frac{1}{1 + e^{2 x}}) = l o g y

- 2 + 4 = l o g y

y = e^{2}

Commented by Dwaipayan Shikari last updated on 07/Sep/20

\lim_{x \to \frac{π}{2}} {(c o s x)}^{- x + \frac{π}{2}} = y

\Rightarrow - (x - \frac{π}{2}) l o g (c o s x) = l o g y

\Rightarrow - (x - \frac{π}{2}) (c o s x - 1) \frac{l o g (1 + c o s x - 1)}{c o s x - 1} = l o g y

\Rightarrow 2 (x - \frac{π}{2}) {s i n}^{2} \frac{x}{2} = l o g y

\Rightarrow l o g y = 0

y = 1

Answered by bemath last updated on 07/Sep/20

Commented by mohammad17 last updated on 07/Sep/20

s i r i n s t e p f o u r (\frac{t a n x + s i n x}{∣ x ∣}) n o t (\frac{t a n x - s i n x}{∣ x ∣})

Commented by bemath last updated on 07/Sep/20

oo yes you are right

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