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

Question

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.

Answer 1

How about this:

$$0<\frac{1}{1+y^2}<1,$$ meaning that $$\frac{1}{1+y^2}e^{-ay} < e^{-ay}$$ and

$$\int _0^\infty\frac{1}{1+y^2}e^{-ay}dy < \int_0^\infty e^{-ay}dy$$