_ x 0 ( 3 x^ 2 +2 7 x^ 2 +2 )^ 1 x^ 2 is equal to - Vidyadip

MCQ

$\lim\limits_{x \rightarrow 0}\left(\frac{3 x^{2}+2}{7 x^{2}+2}\right)^{\frac{1}{x^{2}}}$ is equal to

A
$\frac{1}{e}$
B
$e^2$
C
$e$
✓
$\frac{1}{e^2}$

✓

Answer

Correct option: D.

$\frac{1}{e^2}$

d
Required limit $=e^{\lim _{x \rightarrow 0}\left(\frac{3 x^{2}+2}{7 x^{2}+2}-1\right) \frac{1}{x^{2}}}$

$=e^{\lim _{x \rightarrow 0}\left(\frac{-4}{7 x^{2}+2}\right)}=\frac{1}{e^{2}}$

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