Prove that \[\lim_{x\to 4}\,\sqrt{x}=2.\]

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

Prove that \[\lim_{x\to 4}\,\sqrt{x}=2.\]

Answer: 

It is true that \[\lim_{x\to 4}\,\sqrt{x}=2.\]

SAMPLE SOLUTION

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Solution: 

To prove this we need to show that for every <Sign in to see all the formulas> there exists a <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>. 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 $|x-4|$ and <Sign in to see all the formulas>. We know that

<Sign in to see all the formulas>


This rule allows us to state that

<Sign in to see all the formulas>


So how does $|x-4|$ relate to <Sign in to see all the formulas>? Because <Sign in to see all the formulas> we know that

<Sign in to see all the formulas>


Keep this in mind because it is an important result that will be used below.

We still have not made at educated guess about <Sign in to see all the formulas>. Since we want <Sign in to see all the formulas> and know that <Sign in to see all the formulas> then maybe we should try <Sign in to see all the formulas>.

Before we do this we have to note that the domain of <Sign in to see all the formulas> is only positive numbers. Rule 7 of our rules of absolute values says that <Sign in to see all the formulas> means that <Sign in to see all the formulas> or <Sign in to see all the formulas>. Since $x$ cannot be negative under <Sign in to see all the formulas> then we need to make sure that <Sign in to see all the formulas> so that <Sign in to see all the formulas> and we consider values of $x$ that are strictly positive. So we should try <Sign in to see all the formulas> where $\min$ is the minimum function.

So let's try <Sign in to see all the formulas>. Now let's assume that we are at an $x$ such that <Sign in to see all the formulas>. Now let's do some simple algebra:

<Sign in to see all the formulas>


So we have proven that <Sign in to see all the formulas>. Implicit in the algebra was our understanding that <Sign in to see all the formulas> with our restriction that <Sign in to see all the formulas>.

Our guess of <Sign in to see all the formulas> worked and the proof is complete.