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Introduction to limits

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Question

$Let e denote the base of the natural logarithm. The value of the real number a for which the right hand limit$
$lim_{x \to 0^{+}} \frac{(1 - x)^{\frac{1}{x}} - e^{- 1}}{x^{a}} is equal to a nonzero real number, is$

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Solution

$\begin{array}{l} lim_{x = 0^{+}} \frac{e^{(\frac{\ln (1 - x)}{x})} - \frac{1}{e}}{x^{a}} \\ = lim_{x - 0^{+}} \frac{1}{e} \frac{e^{(1 + \frac{\ln (1 - x)}{x})} - 1}{x^{a}} \\ = \frac{1}{e} lim_{x - 0^{+}} \frac{e^{(1 + \frac{\ln (1 - x)}{x})} - 1}{1 + \frac{\ln (1 - x)}{x}} \frac{1 + \frac{\ln (1 - x)}{x}}{x^{a}} \\ = \frac{1}{e} lim_{x - 0^{+}} \frac{1 + \frac{l n (1 - x)}{x}}{x^{a}} \\ = \frac{1}{e} lim_{x - 0^{+}} \frac{l n (1 - x) + x}{x^{(a + 1)}} \\ = \frac{1}{e} lim_{x - 0^{+}} \frac{(- x - \frac{x^{2}}{2} - \frac{x^{3}}{3} - \dots \dots .) + x}{x^{a + 1}} \end{array}$
$Thus, a = 1$

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Similar Questions

Match the following lists:

Column -I	Column -II
A. If $L = lim_{x \to - 1} \frac{\sqrt[3]{(7 - x)} - 2}{(x + 1)}$ , then 12L =	P. -2
B. If $L = lim_{x \to π / 4} \frac{\tan^{3} x - \tan x}{\cos (x + \frac{π}{4})}$ then -L/4=	Q. 2
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