Prove that $$\lim_{h\to 0}\,\sqrt{x+h}+\sqrt{x}=2\sqrt{x}.$$

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. (Note that in this problem our function is a function of $h$ and not $x$ as $x$ plays the role of a constant. This is only a difference in notation. We could changed our notation for the limit stuff to say <Sign in to see all the formulas>.)

The first step is to make an educated guess about <Sign in to see all the formulas> by finding a relationship between $|h|$ 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 $|h|$ relate to <Sign in to see all the formulas>? Just do the substraction of the <Sign in to see all the formulas> to get <Sign in to see all the formulas>. So $|h|$ and <Sign in to see all the formulas> are related by

<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 both <Sign in to see all the formulas> and <Sign in to see all the formulas> are both postive we can use this rule to show that <Sign in to see all the formulas>. Here is how we make our educated guess about <Sign in to see all the formulas>. We now know that

<Sign in to see all the formulas>

but we want <Sign in to see all the formulas> which means that

<Sign in to see all the formulas>

So if we choose <Sign in to see all the formulas> then it might be less then $|h|$ and thus delta. We know this is somewhat fuzzy, but unfortunately this is how you start to come by a guess to use. Once you have a guess then you need to go through the proof process to see if your guess was right. So let's 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. Since $x$ is a constant here we don't want $x+h$ to become negative in <Sign in to see all the formulas> so we need to make sure that <Sign in to see all the formulas> and to do this we need to make sure that <Sign in to see all the formulas>. Therefore our guess for <Sign in to see all the formulas> should be <Sign in to see all the formulas> where $\min$ is the minimum function.

So let's try <Sign in to see all the formulas>. Remember that $x$ is a constant here because our function is a function of $h$. Now let's assume that we are at an $h$ such that <Sign in to see all the formulas>. Now let's do some simple algebra:

<Sign in to see all the formulas>

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