is equal to [2025]

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Solution

Q.
$\lim_{x \to 0^{+}} \frac{\tan (5 {(x)}^{\frac{1}{3}}) \log_{e} (1 + 3 x^{2})}{{(\tan^{- 1} 3 \sqrt{x})}^{2} (e^{5 {(x)}^{\frac{4}{3}}} - 1)}$ is equal to [2025]

1 1

2 $\frac{5}{3}$

3 $\frac{1}{3}$

4 $\frac{1}{15}$

Ans.
(3)

We have, $\lim_{x \to 0^{+}} \frac{\tan (5 {(x)}^{\frac{1}{3}}) \log_{e} (1 + 3 x^{2})}{{(\tan^{- 1} (3 \sqrt{x}))}^{2} (e^{5 {(x)}^{\frac{4}{3}}} - 1)}$

$\Rightarrow \lim_{x \to 0^{+}} \frac{\frac{\tan (5 {(x)}^{\frac{1}{3}}) \log_{e} (1 + 3 x^{2})}{3 x^{2} \times 5 {(x)}^{\frac{1}{3}}} (3 x^{2}) 5 {(x)}^{\frac{1}{3}}}{\frac{{(\tan^{- 1} (3 \sqrt{x}))}^{2}}{{(3 \sqrt{x})}^{2}} \frac{(e^{5 x^{\frac{4}{3}}} - 1)}{5 x^{\frac{4}{3}}} \times 9 x \times 5 x^{\frac{4}{3}}}$

$\Rightarrow \lim_{x \to 0^{+}} \frac{\frac{\tan (5 {(x)}^{\frac{1}{3}})}{5 {(x)}^{\frac{1}{3}}} \frac{\log_{e} (1 + 3 x^{2})}{3 x^{2}} \times 15 x^{\frac{7}{3}}}{\frac{{(\tan^{- 1} (3 \sqrt{x}))}^{2}}{{(3 \sqrt{x})}^{2}} \frac{(e^{5 {(x)}^{\frac{4}{3}}} - 1)}{5 x^{\frac{4}{3}}} \times 45 x^{\frac{7}{3}}} = \frac{1}{3}$ .

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