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Evaluate $\lim \limits_{x \to 2} \bigg( \large\frac{e^x-e^2}{x-2} \bigg)$

$\begin{array}{1 1}(A)\;e \\(B)\;e^2 \\(C)\;e^3 \\(D)\;e^4 \end{array}$

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1 Answer

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Put $(x-2)=y$
So that when $x \to 2$ then $y \to 0$
$\therefore \lim \limits_{x \to 2} \bigg(\large\frac {e^x-e^2}{x-2} \bigg) $$=\lim \limits_{y=0} \bigg(\large\frac {e^{y+2}-e^2}{y} \bigg) $
$\qquad=\lim \limits_{y \to 0} \bigg \{ e^2. \bigg(\large\frac {e^y-1}{y} \bigg) \bigg \}$
$\qquad=e^2.\lim \limits_{y \to 0} \bigg(\large\frac {e^y-1}{y} \bigg)$
$\qquad=e^2 \times 1$
$\qquad=e^2$
Hence B is the correct answer.
answered Apr 22, 2014 by meena.p
 

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