# An Application of Lebesgue Dominated Convergence Theorem

As an application of the Lebesgue Dominated Convergence Theorem, we would like to evaluate the following limit $$\lim_{n\to\infty}\int\limits_1^{n}\frac{1}{\displaystyle\left(1+\frac{x}{n}\right)^nx^{1/n}}\ \mathrm dx.$$ We realized that to put the limit the inside of the integral we need to find an integrable function $$g(x)$$ such that $$\left|\frac{1}{\displaystyle\left(1+\frac{x}{n}\right)^nx^{1/n}}\right| \leq g(x).$$

We did this. But the boundary of the integral also depends on $$n$$. At this point, we do not know what we need to do.

Thanks in advance for any help.

Hint: Using $$(1+z)^n\ge \binom{n}{2}z^2$$ for $$z>0$$, one has $$(1+\frac{x}{n})^n\ge\frac{n(n-1)}{2}\frac{x^2}{n^2}=\frac12(1-\frac1n)x^2\ge\frac14x^2$$ for $$n\ge2$$. Then define $$f_n(x)=\frac{1}{(1+\frac{x}{n})^nx^{1/n}}\,1_{[0,n]}$$ to estimate $$f_n$$ as $$|f_n(x)|\le\frac{4}{x^2}$$ for $$x\ge1$$ and then one can use LDC.
You have $$\int_1^{n}\frac{1}{(1+\frac{x}{n})^nx^{1/n}}dx =\int_1^{\infty}\frac{1}{(1+\frac{x}{n})^nx^{1/n}}\,1_{[0,n]}\,dx.$$ If your $$g$$ does not depend on $$n$$, you are done.