find-lim-x-0-e-1-sinx-e-tanx-

Question Number 29460 by prof Abdo imad last updated on 08/Feb/18

f i n d {l i m}_{x \to 0} \frac{e^{\sqrt{1 + s i n x}} - e}{t a n x} .

Answered by Cheyboy last updated on 09/Feb/18

\lim_{x \to 0} \frac{e^{\sqrt{1 + \sin x}} - e}{\tan x} = 0

Commented by prof Abdo imad last updated on 22/Feb/18

f o r x - \to 0 w e h a v e 1 + s i n x \sim 1 + x a n d

\sqrt{1 + x} \sim 1 + \frac{x}{2} a n d e^{1 + \frac{x}{2}} \sim e (1 + \frac{x}{2}) \Rightarrow

e^{\sqrt{1 + s i n x}} - e \sim \frac{e x}{2} b u t t a n x \sim x \Rightarrow

{l i m}_{x \to 0} \frac{e^{\sqrt{1 + s i n x}} - e}{t a n x} = \frac{e}{2} .

Answered by sma3l2996 last updated on 09/Feb/18

s i n x \underset{0}{\sim} x \Leftrightarrow 1 + s i n x \underset{0}{\sim} 1 + x

\sqrt{1 + s i n x} \underset{0}{\sim} {(1 + x)}^{1 / 2}

\sqrt{1 + x} \underset{0}{\sim} 1 + \frac{1}{2} x

e^{\sqrt{1 + s i n x}} \underset{0}{\sim} e^{1 + \frac{1}{2} x} = e \times e^{\frac{x}{2}}

e^{x / 2} \underset{0}{\sim} 1 + \frac{x}{4}

e^{\sqrt{1 + s i n x}} - e \underset{0}{\sim} e (1 + \frac{x}{4}) - e = \frac{e}{4} x

t a n x \underset{0}{\sim} x

\underset{x \to 0}{l i m} \frac{e^{\sqrt{1 + s i n x}} - e}{t a n x} \sim \underset{x \to 0}{l i m} \frac{\frac{e}{4} x}{x} = \frac{e}{4}

Commented by prof Abdo imad last updated on 22/Feb/18

f o r u \in v (0) e^{u} \sim 1 + u s o e^{\frac{x}{2}} \sim 1 + \frac{x}{2} .

Commented by sma3l2996 last updated on 22/Feb/18

Y e a h, {y o u}^{'} r e r i g h t