Question

The function $f\left(x\right)=\{\begin{array}{c}\frac{\sin \left(\sqrt[3]{3x}\right)\ln (1+3x)}{{\left(\tan {}^{-1}\sqrt{x}\right)}^2(e^{\left(5\sqrt[3]{3x}\right)}-1)}\\ a,\end{array}\begin{array}{c}x\ne 0\\ x=0\end{array}$ is continuous at $x=0$, then value of $x$ is

• A. $0$
• B. $\frac{3}{5}$
• C. $2$
• D. $\frac{5}{3}$

Answer 1

The correct option is B $\frac{3}{5}$

$\begin{array}{c}f\left(x\right)=\{\begin{array}{c}\frac{\sin \left(x^{\frac{1}{3}}\right)\ln \left(1+3x\right)}{\left({\tan }^{-1}\sqrt{x}\right)\left(e^{5\sqrt[3]{3x}}-1\right)};x\ne 0\\ a;x=0\end{array}\\ {lt}_{x\to 0}\frac{\sin \left(x^{\frac{1}{3}}\right)\ln \left(1+3x\right)}{\frac{{\left({\tan }^{-1}\sqrt{x}\right)}^2}{{\left(\sqrt{x}\right)}^2}\left(\sqrt{x}\right)\frac{\left(e^{5\sqrt[3]{3x}}-1\right)5x^{\frac{1}{3}}}{5x^{\frac{1}{3}}}}\\ {lt}_{x\to 0}\frac{\sin \left(x^{\frac{1}{3}}\right)\ln \left(1+3x\right)}{{\left(\sqrt{x}\right)}^2\left(5x^{\frac{1}{3}}\right)}\\ {lt}_{x\to 0}\frac{\sin \left(x^{\frac{1}{3}}\right)\ln \left(1+3x\right)\left(3x\right)}{x^{\frac{1}{3}}\left(3x\right)\left(5x\right)}=\frac{3}{5}\\ a=\frac{3}{5}=f\left(0\right)\\ Hence,theoptionBisthecorrectanswer.\end{array}$