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Thread: How Do You Prove These Limits?

  1. #1
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    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))
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  2. #2
    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by amm345 View Post
    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]=...
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