BeMath-1-1-1-2-1-3-1-4-1-5-1-6-2-lim-x-0-1-sin-x-1-x-

Question Number 107352 by bemath last updated on 10/Aug/20

⊚ B e M a t h ⊚

(1) 1 - \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} - \frac{1}{\sqrt{4}} + \frac{1}{\sqrt{5}} - \frac{1}{\sqrt{6}} + \dots = ?

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

Answered by Dwaipayan Shikari last updated on 10/Aug/20

2) \lim_{x \to 0} {(1 + sinx)}^{\frac{1}{x}} = y

\frac{1}{x} \log (1 + sinx) = logy

\frac{sinx}{x} \frac{\log (1 + sinx)}{sinx} = logy

\frac{x}{x} . 1 = logy

y = e

Answered by john santu last updated on 10/Aug/20

⋇ JS ⋇

(2) \lim_{x \to 0} {(1 + \sin x)}^{\frac{1}{x}} = e^{\lim_{x \to 0} (1 + \sin x - 1) . \frac{1}{x}}

= e^{\lim_{x \to 0} (\frac{\sin x}{x})} = e^{1} = e

Answered by Dwaipayan Shikari last updated on 10/Aug/20

\underset{n = 1}{\sum^{\infty}} \frac{{(- 1)}^{n + 1}}{\sqrt{n}} = 0.609 \dots

Commented by Dwaipayan Shikari last updated on 10/Aug/20

Answered by mathmax by abdo last updated on 10/Aug/20

let f (x) = {(1 + sinx)}^{\frac{1}{x}} \Rightarrow f (x) = e^{\frac{1}{x} \ln (1 + sinx)}

we have sinx \sim \Rightarrow 1 + sinx \sim 1 + x \Rightarrow \ln (1 + sinx) \sim \ln (1 + x) \sim x (x \to 0)

\Rightarrow \frac{1}{x} \ln (1 + sinx) \sim 1 \Rightarrow \lim_{x \to 0} f (x) = e

Answered by bemath last updated on 10/Aug/20

⊚ B e M a t h ⊚

(2) \lim_{x \to 0} {(1 + \sin x)}^{\frac{1}{x}} = e^{\lim_{x \to 0} \frac{\ln (1 + \sin x)}{x}}

= e^{\lim_{x \to 0} (\frac{\sin x - \frac{\sin^{2} x}{2} + \dots}{x})} = e^{1} = e

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