_ x 0 (1+5 x^2 1+3 x^2 )^ 1 / x^2 = - Vidyadip

MCQ

$\lim _{x \rightarrow 0}\left(\frac{1+5 x^2}{1+3 x^2}\right)^{1 / x^2}=$

✓
$e^2$
B
e
C
$e ^{-2}$
D
$e ^{-1}$

✓

Answer

Correct option: A.

$e^2$

(A)
$\lim _{x \rightarrow 0}\left(\frac{1+5 x^2}{1+3 x^2}\right)^{1 / x^2}=\frac{\lim _{x \rightarrow 0}\left[\left(1+5 x^2\right)^{1 / 5 x^2}\right]^5}{\lim _{x \rightarrow 0}\left[\left(1+3 x^2\right)^{1 / 3 x^2}\right]^7}$
$=\frac{ e ^5}{ e ^3} \quad \ldots .\left[\because \lim _{x \rightarrow 0}(1+x)^{1 / x}= e \right]$
$= e ^2$

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