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Prove that \begin{equation} \lim_{x\to\infty}\,\frac{1}{x^n}=0. \end{equation}

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Lesson Parent:

Problem:

Prove that
\begin{equation}
\lim_{x\to\infty}\,\frac{1}{x^n}=0
\end{equation}
where $n=1,2,3,\cdots$.

Answer:

It is true that \begin{equation} \lim_{x\to\infty}\,\frac{1}{x^n}=0. \end{equation}

Solution:

The proof is by induction. We have already show that this is true for $n=1$ and $n=2$.

Now let's assume that it is true for some arbitrary integer $m$. This means that we are assuming

Given this assumption we must now show that

Notice that

This completes the proof.

Recall that what we are doing is showing that if <Sign in to see all the formulas> and <Sign in to see all the formulas> are true then we can start at some integer value we know the limit is true and be guaranteed that it is true for the next one. More specifically, we know
$m=2$ and have shown that if <Sign in to see all the formulas> and <Sign in to see all the formulas> is true then <Sign in to see all the formulas> is true. Therefore, it is true for $m=3$ and we can say

Therefore,