Explanation

(b)
Given limit
$= \lim_{x \rightarrow 0}\frac{\int_{0}^{x}{\frac{t\log{(1 + t)}}{t^{4} + 4}dt}}{x^{3}}$
Using L’ Hospital’s rule,
$= \lim_{x \rightarrow 0}\frac{\frac{x\log{(1 + x)}}{x^{4} + 4}}{{3x}^{2}}$
$= \lim_{x \rightarrow 0}\ \frac{\log{(1 + x)}}{3x}.\frac{1}{x^{4} + 4}$
$= \frac{1}{3}.\frac{1}{4} = \frac{1}{12}$