lim x→0^+ tan(5(x)^(1/3)) loge(1+3x^2)/(tan^(-1)3√x)^2 (e^(5(x)^(4/3))-1) is equal to

Question

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

(1) \(\frac{1}{15}\)

(2) 1

(3) \(\frac{1}{3}\)

(4) \(\frac{5}{3}\) 

Answer 1

Correct option is: (3) \(\frac{1}{3}\) 

\(\lim\limits _{x \rightarrow 0^{+}}\left(\frac{\tan \left(5 x^{1 / 3}\right)}{5 x^{1 / 3}}\right) \cdot\left(\frac{(3 \sqrt{x})^{2}}{\left(\tan ^{-1} 3 \sqrt{x}\right)^{2}}\right)\left(\frac{\ell\left(1+3 x^{2}\right)}{3 x^{2}}\right)\left(\frac{5 x^{4 / 3}}{5 e^{\frac{4}{3}}-1}\right) \times \frac{5 x^{1 / 3} \cdot 3 x^{2}}{5 x^{4 / 3} \cdot 9 x}\)

\(=\frac{1}{3}\)