Thread: How Do You Prove These Limits?

1. How Do You Prove These Limits?

Can someone please explain how to prove these? Thanks!
Let k,l be natural numbers such that k>l>2.

limit as n tends to inifity of:
a) ((n^k)+1)/(10(n^l)+5(n^2)+n)

b)((n^(k+l))+(n^k)+(n^l))/(2^n)

c)n^(1/sqrt(n))

2. Originally Posted by amm345
Can someone please explain how to prove these? Thanks!
Let k,l be natural numbers such that k>l>2.

limit as n tends to inifity of:
a) ((n^k)+1)/(10(n^l)+5(n^2)+n)

b)((n^(k+l))+(n^k)+(n^l))/(2^n)

c)n^(1/sqrt(n))
a.) $\lim_{n\to\infty}\frac{n^k+1}{10n^l+5n^2+n}=\lim_{ n\to\infty}\frac{n^l(n^{k-l}+n^{-l})}{n^l(10+5n^{2-l}+n^{1-l})}=...$

b.) $\lim_{n\to\infty}\frac{n^{k+l}+n^k+n^l}{2^n}=\lim_ {n\to\infty}\frac{n^{k+l}(1+n^{-l}+n^{-k})}{2^n}=$ $\lim_{n\to\infty}\frac{n^{k+l}}{2^{n/2}}\cdot\lim_{n\to\infty}\frac{1+n^{-l}+n^{-k}}{2^{n/2}}=...$

c.) $\lim_{n\to\infty}n^{1/\sqrt{n}}=\lim_{n\to\infty}\exp\left[\ln\left(n^{1/\sqrt{n}}\right)\right]=\lim_{n\to\infty}\exp\left[\frac{\ln n}{\sqrt{n}}\right]=...$