Proving that a family of functions converges to the Dirac delta.

I think it is easier to do it without complex analysis, just using elementary results on integration theory. The following is a more general, often useful result.

Let $K\in L^1(\mathbf{R})$, for $\varepsilon>0$ let $K_\varepsilon$ be given by $K_\varepsilon(x)=\frac{1}{\varepsilon}K(x/\varepsilon)$; assume also that the integral of $K$ over all of $\mathbf{R}$ is equal to $1$. Then if $\phi\in L^\infty(\mathbf{R})$ we have $\phi_\varepsilon=\phi\ast K_\varepsilon\rightarrow\phi$ for $\varepsilon\downarrow 0$ at every point of continuity of $\phi$. (This is theorem 9.8 in Wheeden's measure and integral.)

Now for your case take the poisson kernel $$K(x)=\frac{1}{\pi}\frac{1}{1+x^2},$$ and $\phi$ be of class $C^\infty_0(\mathbf{R})\subset L^\infty(\mathbf{R})$. Then just notice that $$\phi_\varepsilon(0)=(\phi\ast K_\varepsilon)(0)=\frac{1}{\pi}\int_{\mathbf{R}}\phi(x)\frac{\varepsilon}{\varepsilon^2+x^2}dx.$$

For the proof of the result cited above (which is not that hard and a standard argument anyway):

(As you can see, there is some notational conflict between the notation used in the book and the notation which you use in the question; I suggest we ignore this.)