# Show $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0$

Need to prove $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy =0$ and $\lim\limits_{a \rightarrow + \infty} \int_0^{\infty} \frac{y}{1+y^2}e^{-ay} dy =0$

Can someone solve using dominated convergence theorem? I want to know how LDC is applied.

-
Dominated convergence theorem definitely gives you a single-line proof for the first one. Just observe that the integrands are dominated by an integrable function $(1 + y^{2})^{-1}$. But 5xum's idea is much easier to utilize in both of the problems. –  Sangchul Lee Apr 22 '14 at 9:18
Do you mean I just do Riemann integral? I am confused if Lebesgue is used in this problem. –  user220055 Apr 22 '14 at 9:28
@user220055 For continuous functions, there is no difference. –  5xum Apr 22 '14 at 9:32
Right, thank you –  user220055 Apr 22 '14 at 9:33

$$0<\frac{1}{1+y^2}<1,$$ meaning that $$\frac{1}{1+y^2}e^{-ay} < e^{-ay}$$ and
$$\int _0^\infty\frac{1}{1+y^2}e^{-ay}dy < \int_0^\infty e^{-ay}dy$$
$\lim\limits_{a \rightarrow + \infty} cos (a)\int_0^{\infty} \frac{1}{1+y^2}e^{-ay} dy$ is in another problem. Is it legal to say it is 0? And do I need to prove continuity for the question I asked? –  user220055 Apr 22 '14 at 10:42
If $$\lim_{a\to\infty} f(a) = 0$$ and $g$ is a bounded function, then it is easy to see that $$\lim_{a\to\infty} f(a)g(a) = 0.$$ –  5xum Apr 22 '14 at 10:50