_ x 0^ + (5(x)^ 1 3 ) _ e (1+3 x^ 2 ) ( ^ -1 3 x )^ 2 (e^ 5(x)^ 4… - Vidyadip

MCQ

$\lim _{x \rightarrow 0^{+}} \frac{\tan \left(5(x)^{\frac{1}{3}}\right) \log _{e}\left(1+3 x^{2}\right)}{\left(\tan ^{-1} 3 \sqrt{x}\right)^{2}\left(e^{5(x)^{\frac{4}{3}}}-1\right)}$ is equal to

A
$\frac{1}{15}$
B
1
C
$\frac{1}{3}$
D
$\frac{5}{3}$

✓

Answer

C. $\frac{1}{3}$
$\lim _{x \rightarrow 0^{+}}\left(\frac{\tan \left(5 x^{1 / 3}\right)}{5 x^{1 / 3}}\right) \cdot\left(\frac{(3 \sqrt{x})^{2}}{\left(\tan ^{-1} 3 \sqrt{x}\right)^{2}}\right)\left(\frac{\ell\left(1+3 x^{2}\right)}{3 x^{2}}\right)\left(\frac{5 x^{4 / 3}}{e^{5 x{\frac{4}{3}}}-1}\right) \times \frac{5 x^{1 / 3} \cdot 3 x^{2}}{5 x^{4 / 3} \cdot 9 x}$
$=\frac{1}{3}$

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