The weak law of large numbers and the central limit theorem are among the most popular limit theorems from probability theory. Moreover, both results have widespread applications in the whole of probability and statistics. What is less known is that these results can also provide probabilistic proofs of real inversion formulas from transform theory. We illustrate this statement with classical (and some less classical) cases from the theory of integral transforms, namely the Laplace transform and the Stieltjes transform. We add a few new results as well.