I want to calculate $$\lim_{t\to \infty}\int_0^1 \frac{e^{-tx}}{1+x^2}\ dx.$$
My idea is using the Dominated Convergence Theorem (DCT).
Now, for $x\in [0,1]$, $$\displaystyle\left|\frac{e^{-tx}}{1+x^2}\right|\leqq \frac{1}{1+x^2}$$ and RHS is independent of $t$ and integrable on $[0,1]$.
Thus, if I use DCT, I get $$\lim_{t\to \infty}\int_0^1 \frac{e^{-tx}}{1+x^2}\ dx=\int_0^1 \lim_{t\to \infty}\frac{e^{-tx}}{1+x^2}\ dx=0.$$
But I wonder if I'm able to use DCT.
The outline of DCT is :
If $\{f_n\}_{n=1}^\infty$ is a sequence of measurable functions and $\forall x ; f_n(x)\to f(x)$, and there exists integrable $g(x)$ s.t. $|f_n(x)|\leqq g(x) \ \forall n$, then $\displaystyle\lim_{n\to \infty} \int f_n(x)dx=\int f(x)dx.$
In this case, the variable $t$ is not natural number, so I'm not sure whether I can do $\lim_{t\to \infty}\int f(t,x)dx=\int \lim_{t\to \infty}f(t,x) dx.$
Could you explain for this problem ?
Best Answer
In your case, the dominated convergence theorem applies because for any function $f:[0,\infty)\to\mathbb R$ you have $\lim\limits_{t\to\infty}f(t)=c$ if and only if $\lim\limits_{n\to\infty} f(t_n)=c$ for every sequence $t_n\to\infty$.
For general nets however, DCT may fail. Let $I$ be the irected set of all finite subsets $E\subseteq [0,1]$ and $f_E$ the indicator function of $E$. The net converges pointwise to the indicator function $f$ of $[0,1]$, it is majorized by this $f$ but the integrals are $\int f_E(x)dx=0$ and $\int f(x)dx=1$.