calculus - Computing $limlimits_{xto0}int_0^xfrac{t^2}{(x-sin x)sqrt{a+t}},dt$ without L'Hopital

I am computing the following limit

$$\lim_{x\to0}\int_0^x\frac{t^2}{(x-\sin x)\sqrt{a+t}}\,dt$$
where $a$ is a parameter. the case $a= 0$ has been resolved. But yet, for the case $a\neq 0$ I want to compute it without using L'Hopital rule.

Answer

If you know how to calculate the limit of $\frac{x^3}{x-\sin x}$ without L'Hospital, then the rest follows from the mean value theorem for integrals (if we may use it - it is quite similar to L'Hospital/Bernoulli's rule)

$$\int_0^x\frac{t^2}{x^3\sqrt{a+t}}\,dt=\left[u=\frac{t}{x}\right]=\int_0^1\frac{u^2}{\sqrt{a+xu}}\,du=\frac{1}{\sqrt{a+x\xi}}\int_0^1u^2\,du=\frac{1}{\sqrt{a+x\xi}}\cdot\frac13\to\frac{1}{3\sqrt{a}}.$$