If $f\left(x\right)=\lim_{n\to \infty} n^{2}\left(x^{\frac{1}{n}}-x^{\frac{1}{n+1}}\right)$ , $x>0$ then $\int xf\left(x\right)$ is?

Question

Tried bringing $n^{2}$ to the Denominator and splitting but no luck just give me a clue on how to start approach

Answer 1

Sorry can't actually add a comment so writing here: Putting. $$ n=\frac{1}{t} \Rightarrow \lim_{t\rightarrow 0} \frac{x^{t}-x^{\frac{t}{t+1}}}{t^{2}} $$ now applying l'hospital rule twice to get $$ f(x) = ln(x) $$ Now you can proceed

Answer 2

Hint: $$ x^{1/n} = 1 + \frac{\log x}{n} + \frac{1}{2}\frac{\log^2 x}{n^2} + \ldots $$ for $x>0$. Collect and simplify.