Prove that the lim as x approaches infinity of x^n/n^k=+infinity if x>1
Lets assume that you mean x and not k as the other poster said...then you want to show that
$\displaystyle \lim_{n\to\infty}\frac{x^n}{n^x}=\infty$
Consider that $\displaystyle \frac{d^{x+1}}{dn^{x+1}}\bigg[x^n\bigg]=\ln(x)^{x+1}x^n$
and $\displaystyle \frac{d^{x+1}}{dn^{x+1}}\bigg[n^x\bigg]=x!$
So now consdier that
$\displaystyle \lim_{n\to\infty}\frac{x^n}{n^x}=\frac{\infty}{\in fty}$
Therefore L'hopital's may be applied...but we see that this also gives infinity over infinity...this continues until the x+1th derivative where we see that
$\displaystyle \lim_{n\to\infty}\frac{x^n}{n^x}\underbrace{=\cdot s=}_{x \text{ times}}\lim_{n\to\infty}\frac{\ln(x)^{x+1}x^n}{x!}$
Which clearly equals infinity.