Prove that \begin{equation} \lim_{x\to 0}\,4x+2\neq 6. \end{equation}

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Lesson Parent: 
Function Limits III: The Gory Details
Problem: 

Prove that
\begin{equation}
\lim_{x\to 0}\,4x+2\neq 6.
\end{equation}

Answer: 

It is true that \begin{equation} \lim_{x\to 0}\,4x+2\neq 6 \end{equation}

SAMPLE SOLUTION

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Solution: 

Let's start out by being intuitive about this problem. We know that when $x=0$, <Sign in to see all the formulas> and not $6$. However, for now let's assume that

<Sign in to see all the formulas>


is actually true and see if we can use our function limit steps to prove it. Here are the general set of steps we need to take to do our proof.

Here are the steps for this problem:

Step Description
1 Let's assume that <Sign in to see all the formulas> where $N$ is some integer we will try to find later.
2 Say <Sign in to see all the formulas>.
3 Again, we will try to find the integer that makes all of this work.
4 Now assume that we are at some value for $x$ such that <Sign in to see all the formulas>.
5 Now plug the function <Sign in to see all the formulas> into <Sign in to see all the formulas> so that we have <Sign in to see all the formulas>.
6 Now do some basic algebra to get <Sign in to see all the formulas> into the form <Sign in to see all the formulas>. The value of $D$ will naturally appear when you put <Sign in to see all the formulas> in the form <Sign in to see all the formulas>. Here we go:

<Sign in to see all the formulas>

7 Since the smallest <Sign in to see all the formulas> can get is 1 when $N=0$ we see that our algebra above yields <Sign in to see all the formulas>. But we chose <Sign in to see all the formulas>. This means that I want to find a delta such that when <Sign in to see all the formulas> then <Sign in to see all the formulas>. Let's do some algebra:

<Sign in to see all the formulas>


But the <Sign in to see all the formulas> that I chose can never do better than <Sign in to see all the formulas>. That means for our value of <Sign in to see all the formulas> I can't actually find a value for <Sign in to see all the formulas> that gets me as close to 6 as I want. We must conclude that

<Sign in to see all the formulas>

To say that the function limit exists I must be able to find a <Sign in to see all the formulas> for every <Sign in to see all the formulas>. I have one <Sign in to see all the formulas> for which I cannot find a <Sign in to see all the formulas>. I can't say that the function limit exists. Our intuition about the function limit not being 6 was correct. We would say that as we get closer and closer to 0 we don't get closer and closer to 6.

But since we think that the limit is actually 2 how do we prove it. See this problem for that proof.