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

JEE Main · 2025 · 07 Apr 2025 · Shift 1 · Limits

lim_{x \to 0^{+}} \frac{t a n ( 5 ( x ) ^{\frac{1}{3}} ) l o g _{e} ( 1 + 3 x ^{2} )}{( t a n ^{- 1} 3 x ) ^{2} ( e ^{5 (x)^{\frac{4}{3}}} - 1 )}

is equal to

Answer: (C)

x \to 0^{+}

each factor is replaced by its leading behaviour:

tan (5 x^{1/3}) \sim 5 x^{1/3}

lo g_{e} (1 + 3 x^{2}) \sim 3 x^{2}

(tan^{- 1} 3 x)^{2} \sim (3 x)^{2} = 9 x

, and

e^{5 x^{4/3}} - 1 \sim 5 x^{4/3}

. The expression becomes

\frac{5 x ^{1/3} \cdot 3 x ^{2}}{9 x \cdot 5 x ^{4/3}} = \frac{15 x ^{7/3}}{45 x ^{7/3}} = \frac{1}{3}

. The limit equals

\frac{1}{3}

JEE Main · 2025 · 07 Apr 2025 · Shift 1 · Single correct option

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