I've recently seen a proof of the inversion formula for the Fourier transform for $f,\hat{f}\in L^1(\mathbb{R}^d)$.
The main idea of the proof is this
We have $$\int_{\mathbb{R}^d}e^{2\pi i x\cdot \xi}\hat f(\xi)d\xi=\int_{\mathbb{R}^d}e^{2\pi i x\cdot \xi}\left(\int_{\mathbb{R}^d}e^{-2 \pi i y\cdot \xi}dy\right)d\xi.$$ But at this point Fubini doesn't apply thus we can't change the two integrals.
Nevertheless, here is the trick (which I find really smart) for $t>0$ we consider this modified version of the integral $$I_t(x):=\int_{\mathbb{R}^d}\hat f(\xi)e^{-\pi t^2\vert\xi\vert^2}e^{2\pi i x\cdot\xi}d\xi.$$ Now we can use Fubini and compute the integral in two ways and for $t\rightarrow0$ we get the result. By the way everything works because the damping factor is an eigenvector of the Fourier transform of eigenvalue 1 (so it is far from being a random choice).
I was astonished by the proof, we considered an apparently more complicated expression which allowed us to use a particular property and then passed to the limit and obtained the result.
My question is: what are other examples of the same idea?
I know that basically anything which involves approximation with smooth functions falls in this class but I'm asking for some particular example/application (of this or a similar idea) that made you think 'cool'.
(details are welcome).
Computing real integrals via complex contour integration and limits seems to fit exactly.
This also happens in discrete mathematics: I recently came to a combinatorial proof of the identity $$ \sum_{k=0}^{n}{(-1)^k\binom{n}{k}\frac{1}{2k+1}}={(2n)!!\over (2n+1)!!} $$ where the key is to consider the properties of $\sum_{k=0}^{n}{(-1)^k\binom{n}{k}\frac{1}{2k+m}}$ for arbitrary values of $m$.