Limits by Rationalization: An Introduction to Calculus

Sorry for the break in content, everyone. It’s been a stressful few weeks for me.

In our previous article, we did some more practice with limits with factorization, the first tool we can use to algebraically find limits.

Let’s add to these tools, and learn another way to compute limits.

Take, for example:

As always, when faced with a difficult problem, it’s best to take a step back, and try the methods you already know.

See if you can determine the limit graphically, and maybe you can start to picture how we can find it algebraically:

Obviously, this function cannot be factored, and using direct substitution would lead to the indeterminate form. So how can we solve this?

Remember the difference of squares? It was one of the steps used in the previous problem.

Recall that the general formula for the difference of squares is:

This was an essential part of Algebra, and it’s still essential now.

Still treating the limit like a regular function, did you notice that the bottom terms with the square root can be rewritten via a difference of squares?

How, you ask?

Treat the square root of x+5 as one term, and treat 2 as one term.

Let’s allow the square root of x+5 to be equal to a, and let’s allow 2 to be equal to b.

All together, our new denominator would read: a-b

Look familiar yet? We can then multiply by a+b, to get (a²-b²):

How is this useful?

Well, what is the square root of (x+5) squared? It’s just x+5.

The square root and the square cancel out. By using this method, we have gotten rid of the square root.

2 squared is simply 4.

All together, x+5–4 is equal to x+1, so by using the difference of squares, we have reduced the complexity of the denominator greatly.

But wait! Since you multiplied by (a+b) in the denominator, you also have to multiply by (a+b) in the numerator, otherwise the fraction wouldn’t be equal!

Yes, exactly. Thus, we have the equation:

With our current repertoire of tools, we can then evaluate the limit. We can start by eliminating (x+1):

And finish by plugging in (-1):

And we get 4 as our final answer.

This technique is known as “limits by rationalization”, and it solves limits that factorization alone cannot.

With these tools, solving limits algebraically becomes much easier. Let’s continue to build on and practice these tools in the next article.

This is the seventh installation in a series of articles attempting to explain the entirety of Calculus AB to a mainstream audience. It is designed to be easily understood, without resorting to dumbing down complex and nuanced topics. A full article list can be found on my profile.