_ x 0 (3+5 x 3-4 x )^ 1 x = - Vidyadip

MCQ

$\lim _{x \rightarrow 0}\left(\frac{3+5 x}{3-4 x}\right)^{\frac{1}{x}}=$

✓
$e^3$
B
$e^6$
C
$e^9$
D
$e ^{-3}$

✓

Answer

Correct option: A.

$e^3$

(A)
$\lim _{x \rightarrow 0}\left(\frac{3+5 x}{3-4 x}\right)^{\frac{1}{x}}=\lim _{x \rightarrow 0}\left(\frac{1+\frac{5 x}{3}}{1-\frac{4 x}{3}}\right)^{\frac{1}{x}}$
$=\frac{\lim _{x \rightarrow 0}\left[\left(1+\frac{5 x}{3}\right)^{\frac{3}{5 x}}\right]^{\frac{5}{3}}}{\lim _{x \rightarrow 0}\left[\left(1-\frac{4 x}{3}\right)^{\frac{-3}{4 x}}\right]^{\frac{-4}{3}}}$
$=\frac{ e ^{\frac{5}{3}}}{ e ^{\frac{-4}{3}}}= e ^3$

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