TITLE: existence of a limit in a improper integral
QUESTION [1 upvotes]: Consider the limit 
  $$ \lim_{n \to \infty} \int_0^n e^{-px} \cos(ax)\, dx.$$
The exercise is asking about the values of $p$ and $a$ which the limit above exists.
a) exists for infinity values of $p$
b) exists for a finite values of $a$
c) exists for every real $p$ but not for every value of $a$
d) exists for every real $p$ and real $a$
e) none of the above

After a while I'm guessing the answer d) is correct. But I am not sure. Also, asking if the limit above exists, is the same thing as asking if the integral converges? If I evaluate the limit for a negative $p$ and a positive $a$ the integral oscillates between -infinity and infinity. That means the integral diverges and the limit above doesn't exist? I will be thankful if someone can provide a answer and a explanation.

REPLY [2 votes]: Well, notice we are looking at the Laplace transform:
$$\mathscr{L}_x\left[\cos\left(\text{a}\cdot x\right)\right]_{\space\left(\text{p}\right)}:=\int_0^\infty\cos\left(\text{a}\cdot x\right)\cdot e^{-\text{p}t}\space\text{d}t=\frac{\text{p}}{\text{p}^2+\text{a}^2}\tag1$$
When $\Re\left(\text{p}\right)>0$
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