$ \lim_{c \to a} \int_0^c f(x,c) \space dx $

Question

Im practicing limits and integral transforms but I run into the following general problem

How to compute the following limit :

$$ \lim_{c \to a} \int_0^c f(x,c) \space dx $$

Ofcourse this reduces to a normal integral when the limit exists and $a$ is not $0$ nor $\infty$. But when $a$ is one of those two, Im not certain How to proceed.

Can l’hopital be used and justified here ?

Ofcourse most $f$ might be easy, but the hard ones or a general way require Some insights.

I guess $f(x,c) = g(x) \space \ln(1+ c -x) \space (c-x)^n $ might be a hard case.

I think I may have seen those type in analytic Number theory.

Answer 1

I think you mean $$\lim_{c\to a}\int^c_0f(x,c)dx$$

For these type of limits, a typical approach is:

Let $u=\frac{x}c$.

Then the limit becomes $$\lim_{c\to a}\int^1_0cf(cu,c)du$$

Afterwards, you may try to justify the exchange of integral and limit (there are some useful theorems like monotone convergence theorem and dominated convergence theorem).

Suppose the limit exists and $g(u)=\lim_{c\to a}cf(cu,c)$. The problem thus is transformed into a simple integration problem: $$\int^1_0 g(u)du$$

Answer 2

$x$ is just a dummy integration variable so the definite integral doesn't depend on $x$, and the limit doesn't make any sense.