Prove that the limit \begin{equation} \lim_{x\to\infty}\,ax^n \end{equation} does not exist for all $n\geq 1$.
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Prove that the limit
\begin{equation}
\lim_{x\to\infty}\,ax^n
\end{equation}
does not exist for all $n\geq 1$.
SAMPLE SOLUTION
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Let's prove for an arbitrary number $K$ that the limit
<Sign in to see all the formulas>
does not exist.
Recall our formal definition of a limit. If the limit exists then for every <Sign in to see all the formulas> there exists a number <Sign in to see all the formulas> such that if <Sign in to see all the formulas> then
<Sign in to see all the formulas>
To show that the limit does not exist we need prove the logical denial of this statement. The denial states that there exists an <Sign in to see all the formulas> such that for all <Sign in to see all the formulas> there exists an $x$ where <Sign in to see all the formulas> and
<Sign in to see all the formulas>
When a math problem says that "there exists" we have to find something. First, we have to find the correct <Sign in to see all the formulas>. This particular problem is easy and we will see in a second that it doesn't matter what number we actually choose. We will continue to denote that number by <Sign in to see all the formulas>.
Now for every value of $Y$ we have to show there exists an $x$ where <Sign in to see all the formulas> and <Sign in to see all the formulas>. Let's choose <Sign in to see all the formulas>. Since both <Sign in to see all the formulas> and $|K/a|$ are greater than zero then <Sign in to see all the formulas>. We also see that
<Sign in to see all the formulas>
where <Sign in to see all the formulas>. Since <Sign in to see all the formulas>, <Sign in to see all the formulas>, and <Sign in to see all the formulas> then <Sign in to see all the formulas>. Let's examine the following four cases:
So no matter the value of $a$ or $K$, we have shown that <Sign in to see all the formulas>.
Note that we did not specify the actual number <Sign in to see all the formulas>. Our proof above was good for any value of <Sign in to see all the formulas>! For many mathematical proofs this is not the case. However, this problem is simple and we could do it.
So what we have done is to prove that
<Sign in to see all the formulas>
is not true. But because we left $K$ as an arbitrary quantity the proof was general enough that we could put in any actual number for $K$ and show that
<Sign in to see all the formulas>
is not true.Therefore, there is no number $K$ for which
<Sign in to see all the formulas>
is true. This completes the proof that the limit does not exist.
