Applying Lebesgue Dominated Convergence Theorem

Question

I am looking to prove teh following ,

if $ f \in L^1$ then $\lim_{k\to\infty} \int f(x)e^{-|x|^2/k}dx = \int f(x)dx $

I am looking to see if I can use a susbtituion that will make it easy for me to apply LDCT or whether there is a way to apply LDCT directly.

Answer 1

Note that $|f(x)e^{-|x|^{2}/k}| \leq |f(x)|$ for all $x$ and all $k$. Note that $f$ is integrable by assumption; so $|f|$ is integrable (Try checking the other assumption of LDCT yourself!). Then LDCT gives the desired result.