Evaluate : $\displaystyle\lim_{k\to\infty}\int_0^\infty {1\over1+kx^{10}} \, dx$.
From reading other answer to similar questions, I realized that I may have to use dominated convergence theorem to interchange the limit and integral. It would be nice if someone could just explain how to use that theorem in this context to solve this or if that guess is wrong, just tell the right approach.
You don't need the dominated convergence theorem in this case.
Do a substitution $u=x\cdot k^{1/10}$ to see $$\int_0^\infty \frac{dx}{1+kx^{10}} = k^{-1/10} \int_0^\infty \frac{dx}{1+x^{10}}.$$
So the limit evaluates to $0$.