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Looking for a reference to this question I have realized that there are important applications of Lagrange's inversion formula in asymptotical analysis (although its role in the implicit function theorems is already noted by Andres Caicedo). N.G. de Bruijn's Asymptotic Methods in Analysis, Section 2.3, gives some explicit examples: for instance, one can use the Lagrange inversion for computing the asymptotics of the positive root of the equation $xe^x=1/t$ as $t\to\infty$.