Lim_ x 0 e-(1+2 x)^ 1 2 x x is equal to : - Vidyadip

MCQ

$\operatorname{Lim}_{x \rightarrow 0} \frac{e-(1+2 x)^{\frac{1}{2 x}}}{x}$ is equal to :

✓
$e$
B
$\frac{-2}{\mathrm{e}}$
C
$0$
D
$e-e^2$

✓

Answer

Correct option: A.

$e$

a
$ \operatorname{Lim}_{x \rightarrow 0} \frac{e-e^{\frac{1}{2 x} \ln (1+2 x)}}{x} $

$ =\operatorname{Lim}_{x \rightarrow 0}(-e) \frac{\left(e^{\frac{\ln (1+2 x)}{2 x}-1}-1\right)}{x} $

$ =\operatorname{Lim}_{x \rightarrow 0}(-e) \frac{\ln (1+2 x)-2 x}{2 x^2} $

$ =(-e) \times(-1) \frac{4}{2 \times 2}=e$

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