Q36 / 75 · JEE Main 2025 (7 Apr Shift 1) · Mathematics

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

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

Solution

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

Hence, the correct answer is option: 'C'.

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