Mathematics - Limits Question with Solution | TestHub

${\lim }_{x\to 0}\frac{1}{x^3}{\int }_0^{xt}\frac{\log (1+t)}{4+t^4}dt$ Using L' Hospital's rule, $$\begin{array}{rl}& =\underset{x\to 0}{\lim }\frac{\frac{x\log (1+x)}{4+x^4}}{3x^2}\\ =& \underset{x\to 0}{\lim }\frac{\log (1+x)}{3x}\cdot \frac{1}{4+x^4}\\ =& \frac{1}{3}\cdot \frac{1}{4}=\frac{1}{12}\left[\text{ using, }\underset{x\to 0}{\lim }\frac{\log (1+x)}{x}=1\right]\end{array}$$