is equal to [2025]

Question

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

Smart Achievers · Accepted Answer

(3)

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

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

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

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 $\frac{1}{15}$

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Solution

Q. limx→0+tan (5(x)13)loge(1+3x2)(tan–13x)2(e5(x)43–1) is equal to [2025]

1 1

2 53

3 13

4 115

Ans. (3) We have, limx→0+tan (5(x)13)loge(1+3x2)(tan–1(3x))2(e5(x)43–1) ⇒ limx→0+tan (5(x)13) loge(1+3x2)3x2×5(x)13(3x2)5(x)13(tan–1(3x))2(3x)2(e5x43–1)5x43×9x×5x43 ⇒ limx→0+tan (5(x)13)5(x)13loge(1+3x2)3x2×15x73(tan–1(3x))2(3x)2(e5(x)43–1)5x43×45x73=13.

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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]

2 $\frac{5}{3}$

3 $\frac{1}{3}$

4 $\frac{1}{15}$