Prove that the lim as x approaches infinity of x^n/n^k=+infinity if x>1

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- Oct 29th 2008, 06:52 PMSnooks02Proof with limits
Prove that the lim as x approaches infinity of x^n/n^k=+infinity if x>1

- Oct 29th 2008, 10:50 PMchiph588@
do you mean x, not k?

- Oct 30th 2008, 12:15 PMMathstud28
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.