Question

$\underset{x\to 0}{lim}\frac{\log \left(1+x^3\right)}{{\sin }^{3}x}$ is equal to

Answer 1

$\underset{x\to 0}{lim}\frac{log(1+x^3)}{{\sin }^{3}x}=\underset{x\to 0}{lim}\frac{log(1+x^3)}{x^3}\times \frac{x^3}{{\sin }^{3}x}$

$=\underset{x\to 0}{lim}\frac{d}{dx}\frac{log(1+x^3)}{\frac{d}{dx}{x}^3}\times 1$

$=\underset{x\to 0}{lim}\frac{\frac{1}{1+x^3}\left(3x^2\right)}{3x^2}\times 1$

$=1\times 1=2$.