Let f: R R be defined as f(x)= ll x^ 3 (1- 2 x)^ 2 _ e ( 1+2 x e^… - Vidyadip

MCQ

Let $f: R \rightarrow R$ be defined as

$f(x)=\left\{\begin{array}{ll}\frac{x^{3}}{(1-\cos 2 x)^{2}} \log _{e}\left(\frac{1+2 x e^{-2 x}}{\left(1-x e^{-x}\right)^{2}}\right), & x \neq 0 \\ \,\alpha & , x=0\end{array}\right.$ If $\mathrm{f}$ is continuous at $\mathrm{x}=0$, then $\alpha$ is equal to :

✓
$1$
B
$0$
C
$3$
D
$2$

✓

Answer

Correct option: A.

$1$

a
For continuity

$\lim _{x \rightarrow 0} \frac{x^{3}}{4 \sin ^{4} x}\left(\ln \left(1+2 e^{-2 x}\right)-2 \ln \left(1-x e^{-x}\right)\right)$

$=\alpha$

$\lim _{x \rightarrow 0} \frac{1}{4 x}\left[2 x e^{-2 x}+2 x e^{-x}\right]=\alpha$

$=\frac{1}{4}(4)=\alpha=1$

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