An Application of Lebesgue Dominated Convergence Theorem
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.