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Prove that the limit $\begin{equation} \lim_{x\to\infty}\,ax^n \end{equation}$ does not exist for all $n\geq 1$ .

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Lesson Parent:

Function Limits IX: Horizontal Asymptotes And The Limit Of

$f(x)$ As

$x$ Tends To Infinity

Problem:

Prove that the limit
$\begin{equation} \lim_{x\to\infty}\,ax^n \end{equation}$
does not exist for all $n\geq 1$ .

Solution:

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:

<Sign in to see all the formulas>, <Sign in to see all the formulas>	<Sign in to see all the formulas>
<Sign in to see all the formulas>, <Sign in to see all the formulas>	<Sign in to see all the formulas>
<Sign in to see all the formulas>, <Sign in to see all the formulas>	<Sign in to see all the formulas>
<Sign in to see all the formulas>, <Sign in to see all the formulas>	<Sign in to see all the formulas>

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.

Prove that the limit lim\begin{equation} \lim_{x\to\infty}\,ax^n \end{equation} does not exist for all n\geq 1n\geq 1.

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SAMPLE SOLUTION

GAIN AN ADVANTAGE

Prove that the limit $\begin{equation} \lim_{x\to\infty}\,ax^n \end{equation}$ does not exist for all $n\geq 1$ .