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Computing the limit of an integral of a function series

Mathematics Asked by E2R0NS on December 16, 2020

I am trying to figure out how to compute the limit

$$lim_{t rightarrow infty} int_{-1}^1 frac{cos^2(t^3x^{10})}{tx^2 + 1} , dx.$$

If I exchange the limit and integral (when is this allowed?) then I get the integral of 0. This seems too easy. What should I be looking for?

2 Answers

Note that

$$left|int_{-1}^1 frac{cos^2(t^3x^{10})}{tx^2 + 1} , dxright|leqslantint_{-1}^1 frac{|cos^2(t^3x^{10})|}{tx^2 + 1} , dx leqslant 2 int_0^1frac{1}{tx^2 + 1} , dx = frac{2}{sqrt{t}}int_0^{sqrt{t}} frac{du}{1+u^2}$$

Try to finish from here.

Correct answer by RRL on December 16, 2020

Exchanging integral and limit is allowed here: Your integrand is bounded by 1 so you can use dominated convergence theorem.

Answered by Gono on December 16, 2020

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