Solving $\lim_{t\rightarrow 0}\frac{1}{\pi}\int_{-1}^1 f(x) \frac{t}{t^2+x^2} dx =f(0)$ without dominated convergence
$f \in C[-1,1]$, Prove ${\lim_{h \to 0^+}}{\int_{-1}^1 \frac{h}{h^2+x^2}f(x)\,dx} = \pi f(0)$