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Prove that $\begin{equation}\lim_{x\to a}\,x^2=a^2.\end{equation}$

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Lesson Parent:

Function Limits III: The Gory Details

Problem:

Prove that $\begin{equation}\lim_{x\to a}\,x^2=a^2.\end{equation}$

Answer:

It is true that $\begin{equation}\lim_{x\to a}\,x^2=a^2.\end{equation}$

Solution:

We know that because

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and

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then

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By choosing <Sign in to see all the formulas> we immediately see that

<Sign in to see all the formulas>

While the argument is simple and true, it is not very instructive. Here are the general set of steps we need to take to do our proof. The first step is to make an educated guess about <Sign in to see all the formulas> by finding a relationship between $|x-a|$ and <Sign in to see all the formulas>. We can use this rule to show that

<Sign in to see all the formulas>

To carry on we need to get a sense of the size of

$|x+a|$ . When <Sign in to see all the formulas> then <Sign in to see all the formulas>. This suggests that <Sign in to see all the formulas>.

Now here are the steps of the proof for this particular problem:

Choose <Sign in to see all the formulas>.
Pick a positive number $C$ for <Sign in to see all the formulas>.
<Sign in to see all the formulas>.
Assume that we are at some value of $x$ such that <Sign in to see all the formulas>.
Now <Sign in to see all the formulas>.
Now we find by multiplying both sides by $|x+a|$ that

<Sign in to see all the formulas>

By making the assumption that <Sign in to see all the formulas> we have both <Sign in to see all the formulas> and <Sign in to see all the formulas>. However, in the steps above we did both separately.

So we have now proven that

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in two different ways.

Finally, this means that $x^2$ is a continuous function.