Prove that the limit \begin{equation} \lim_{x\to\infty}\,x \end{equation} does not exist.

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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}\,x
\end{equation}
does not exist.

Answer: 

The limit \begin{equation} \lim_{x\to\infty}\,x \end{equation} does not exist.

SAMPLE SOLUTION

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Solution: 

Let's prove for an arbitrary number $K$ that the limit

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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 as <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 $|K|$ are greater than zero then <Sign in to see all the formulas>. We also see that <Sign in to see all the formulas>. Now when <Sign in to see all the formulas>, <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> we have <Sign in to see all the formulas>. When <Sign in to see all the formulas>, <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

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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

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is not true.

Therefore, there is no number $K$ for which

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is true. This completes the proof that the limit does not exist.