calculus - Prove that $lim_{xrightarrow 1} frac{int_0^xg(t)dt-int_0^1g(t)dt-int_0^1f(t)dt(x-1)}{(x-1)^2=frac{f(1}{2}}$

Prove that

$$\lim_{x\rightarrow 1} \frac{\int_0^xg(t)dt-\int_0^1g(t)dt-\int_0^1f(t)dt(x-1)}{(x-1)^2}=\frac{f(1}{2}$$

Now I now this is a limit of the form $\frac{"0"}{"0"}$ which means I can use L'hopital along with the fundamental theorem of calculus. This is the first time that I've done something with two variable, t and x. Which one am I differentiating the terms for in this case?

Answer

The limit says $x \to 1$, so take the derivative with respect to $x$. Note that the $t$ variables are all variables that inside integrals, so taking the derivative with respect to $t$ doesn't make sense.