MCQ
$\lim_{x \rightarrow 0} \frac{\tan \sqrt[3]{x}\log \left( 1+3x \right)}{{{\left( {{\tan }^{-1}}\sqrt{x} \right)}^{2}}\left( {{e}^{5\sqrt[3]{x}}}-1 \right)}=.......$
- A$\frac{\sqrt{3}}{5}$
- B$1$
- ✓$\frac{3}{5}$
- D$0$
$\lim_{x \rightarrow 0} \frac{\tan \sqrt[3]{x}\log \left( 1+3x \right)}{{{\left( {{\tan }^{-1}}\sqrt{x} \right)}^{2}}\left( {{e}^{5\sqrt[3]{x}}}-1 \right)}$
$=\frac{\lim_{x \rightarrow 0} \frac{tan\sqrt[3]{x}}{\sqrt[3]{x}} \times \sqrt[3]{x} \times \lim_{x \rightarrow 0} \frac{log(1+3x)}{3x} \times 3x}{\lim_{x \rightarrow 0 \ \ \ (\frac{tan^{-1}\sqrt{x}}{\sqrt{x}})^2 \times \lim_{x \rightarrow 0 }\frac{e^{5\sqrt[3]{x}}}{5\sqrt[3]{x}} \times {\sqrt[5]{x}}}}$
$=\lim_{x \rightarrow 0} \frac{\sqrt[3]{x} \times 3x}{5x\sqrt[3]{x}}$
$=\frac{3}{5}$
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