The correct option is A e lim_x→∞(1+1/x)^x At x=0, the value of the given expression is 1^∞ form. We can re write as lim_x→∞e^xlog_e(1+1/x) Expansion of log_e ...

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Byju's Answer

Standard XI

Mathematics

Middle Terms

lim x →∞1+1/x...

Question

$lim x \to \infty (1 + \frac{1}{x})^{x} =$

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Solution

The correct option is A
e

$lim x \to \infty (1 + \frac{1}{x})^{x}$

At x=0, the value of the given expression is $1^{\infty}$ form.

We can re-write as

$lim x \to \infty e^{x l o g_{e} (1 + \frac{1}{x})}$

Expansion of $l o g_{e} (1 + x) = x - \frac{x^{2}}{2} + \frac{x^{3}}{3} . . . . . . .$

$\Rightarrow {lim}_{x \to \infty} e^{x [\frac{1}{x} - \frac{1}{2 x^{2}} + \frac{1}{3 x^{3}} - . . . . .]}$

$= lim x \to \infty e^{[1 - \frac{1}{2 x} + \frac{1}{3 x^{3}} - . . . . .]}$

$p u t x = \infty$

We have $e^{[1 - 0....]}$

$= e$

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limx→∞(1+1x)x=

The correct option is A e limx→∞(1+1x)x At x=0, the value of the given expression is 1∞ form. We can re-write as limx→∞exloge(1+1x) Expansion of loge(1+x)=x−x22+x33....... ⇒limx→∞ex[1x−12x2+13x3−.....] =limx→∞e[1−12x+13x3−.....] put x=∞ We have e[1−0....] =e

$lim x \to \infty (1 + \frac{1}{x})^{x} =$