Evaluating $\displaystyle\lim_{x\space\to\space0} \frac{1}{x^5}\int_0^{x} \frac{t^3\ln(1-t)}{t^4 + 4}\,dt$ 
Evaluate the following limit: $$\lim_{x\space\to\space0} \frac{1}{x^5}\int_0^{x}
 \frac{t^3\ln(1-t)}{t^4 + 4}\,dt$$

Any advice on how to tackle this problem ?

Substituting $t =  xu$, we get
$$\int_0^x dt \frac{t^3 \log{(1-t)}}{t^4+4} = x^4 \int_0^1 du \frac{u^3 \log{(1-x u)}}{4+x^4 u^4} $$
so the limit is,
$$\lim_{x\to 0} \frac1{x}\int_0^1 du \frac{u^3 \log{(1-x u)}}{4+x^4 u^4} =  \lim_{x\to 0} \frac1{x}\int_0^1 du \frac{-x u^4}{4} = -\frac1{20}$$
