Question on using L' Hopital's rule

Question

How to use L' Hopital's to solve the following limit

$$\lim_\limits{K\to\infty} K^s\left(1-\frac x{2r}\right)^{K-s} $$ where $s\in Z^+$ is constant. $r\in Z^+$ is constant. $0<x<1$ is constant

Answer 1

Let consider

$$K^s \left(1-\frac x{2r}\right)^{K-s}=e^{s\log K+(K-s)\log \left(1-\frac x{2r}\right)}\to e^{-\infty}=0$$

indeed

$$s\log K+(K-s)\log \left(1-\frac x{2r}\right)=K\cdot\left( s\frac{\log K}{K}+\frac{(K-s)}{K}\log \left(1-\frac x{2r}\right)\right)\to -\infty$$

Answer 2

Set $$ 1-\frac{x}{2r}=\frac{1}{y^s} $$ and note that $y>1$. Then your limit is $$ \lim_{K\to\infty}\left(\frac{K}{y^{K}}\right)^sy^{s^2} $$ Prove now that $$ \lim_{K\to\infty}\frac{K}{y^K}=0 $$ (with l'Hôpital, if you so wish, there are other methods).