To show $\lim_{x\rightarrow 0} x\ln x$ by using DCT?

Question

I know $\lim_{x\rightarrow 0} x\log x＝０$ can be proved by using l'Hospital, but I heard that this statement can also be shown by using dominated convergence theorem. Hint says we use the function $f（t,x）＝1[x,1]（t）x/t$. Could you give me the process of applying DCT to this problem? Thank you in advance.

Answer 1

$\int_0^{1} I_{(x,1)} (t) \frac x tdt \to \int_0^{1} dt=0$ by DCT because $I_{(x,1)} (t) \frac x t \to 0$ as $x \to 0+$ for every $t$ and $0 \leq I_{(x,1)} (t) \frac x t \leq 1$. The constant function $1$ is integrable on $[0,1]$ so DCT is applicable. Of course, $\int_0^{1} I_{(x,1)} (t) \frac x tdt = -x \ln x$ so we get $x \ln x \to 0$ as $x \to 0+$

Answer 2

Note that for $0<x<1$ $$|x\log x| = \left|\int_x^1\frac{x}{t} \, {\rm d}t\right| \leq \int_x^1\frac{x}{t^{3/2}} \, {\rm d}t = 2\sqrt{x}(1-\sqrt{x}) \, .$$