Find the limit of $\ x_n^{\ a_n} - x_n^{\ b_n}$?

Question

Let $\ a_n$ and $\ b_n$ be two sequences of real numbers so that $a_n\to\infty$ as $n\to\infty$ and $\lim_{n\to\infty}\dfrac{a_n}{b_n}=0$. Do there exist sequences like $x_n$ that take values between $0$ and $1$ so that the limit of $x_n$ is $1$ and $$\lim_{n\to\infty}(x_n^{a_n} - x_n^{b_n})=1?$$

Answer 1

Yes; consider $x_n=1-\frac1n$, $a_n=\log n$, and $b_n=n^2$. Use the fact that $$\lim_{n\to\infty}\left(1+\frac{x}{n}\right)^n=e^x$$to make the conclusion.