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My professor posed a question to us last week about the limit of a function as $k \to \infty$. He asked us to prove that $\int \lim_{k \to \infty} f(x)e^{-\frac{x^2}{k}} dx = \int f(x)dx$. This seems fairly basic, since it can be directly shown that the exponential part of the integrand, $e^{-\frac{x^2}{k}} \to 1$ as $k \to \infty$. Is there something I am missing? Are there extra steps needed to show that the integral of the limit is equal to the limit of the integral? Is this even necessary? I feel like this question is too simple relative to the rest of the material in class, but I'm not sure what I am missing.

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    well, $f=g$ implies $\int f=\int g$...2012-11-20
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    Are you sure he didn't ask $\lim_{k\rightarrow \infty}\int f(x)e^{-x^2/k}\,dx$? In that case, you need to justify switching the integral and the limit.2012-12-03

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