My task is formulated as follows: $$\lim_{x \rightarrow 0}\int_{0}^{x} \frac{cos(t^{3})}{t+x} dt$$
Solution of that limit here: Limit under integral problem.
But I'm searching general tips how to find such limits: $$\lim_{x \rightarrow c}\int_{f(x)}^{g(x)} h(x,t) dt$$
Here we can't differentiate our integral or simply apply l'Hopital's rule. And that task seems without having any general techniques to get solution.
Best Answer
You don't have to apply L'Hopital's Rule. By MVT for integrals, there is $c$ between $0$ and $x$ such that $$ \int_{0}^{x} \frac{\cos(t^{3})}{t+x} dt=\cos(c^3)\int_{0}^{x} \frac{1}{t+x} dt=\cos(c^3)\ln2 $$ and hence $$ \lim_{x\to0}\int_{0}^{x} \frac{\cos(t^{3})}{t+x} dt=\lim_{x\to0}\cos(c^3)\ln2=\ln2. $$