Question

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

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

Answer 1

The correct option is D $a=\frac{3}{5}$

$f\left(x\right)\{\begin{array}{c}\frac{\left(x{)}^{1/3}\log \right(1+3x)}{\left({\tan }^{-1}\sqrt{x}{)}^2\right(e^{5x^{1/3}-1)}},x\ne 0\\ a,x=0\end{array}$

$\therefore$ f(x) is continuous at $x=0$

then, $f\left(0\right)=\underset{x\to 0}{lim}f\left(x\right)$

$a=\underset{x\to 0}{lim}\frac{\sin x^{1/3}log(x+3x)}{\left({\tan }^{-1}\sqrt{x}{)}^2\right(e^{sx1/3-1})}$

$a=\underset{x\to 0}{lim}\frac{x^{\frac{1}{3}\frac{\sin x^{\frac{1}{3}}}{x^{\frac{1}{3}}}\times \frac{\log (1+3x)}{3x}\times 3x}}{x{\left(\frac{{\tan }^{-1}\sqrt{x}}{\sqrt{x}}\right)}^2\times \left(\frac{e^{s{x}^{\frac{1}{3}-1}}}{s{x}^{\frac{1}{3}}}\right)s{x}^{\frac{1}{3}}}$

$a=\frac{\perp .\perp .3}{\perp .\perp .5}=\frac{3}{5}$

$a=\frac{3}{5}$

So, option D is correct.