Using Factorization to Find Limits: An Introduction to Calculus

In our previous article, we discovered that limits can be found algebraically via direct substitution, and provided some intuition for why that is.

Let’s remind ourselves of how to find limits via direct substitution.

For a function such as:

We can plug in 1 for x, and reach our answer of 48.

Knowing that, let’s try another problem:

Does the limit as x→5 exist? (Hint: Yes, it does)

Let’s plug in 5 wherever we see x:

Uh oh. 0 over 0 is undefined, so we can’t simplify further.

So does the limit not exist? That can’t be true, because we’ve already established that the limit does exist. So what’s going on here?

The answer lies in that at x=5, there exists a hole, a type of discontinuity.

Why is it a discontinuity? Well, f(5) is undefined. You can’t draw the function without picking up your pencil at x=5.

We said earlier that we can only find the limit of a function using direct substitution if it is continuous. So do we need a new way of finding the limit? Kind of.

Notice that the (x-5) term can be eliminated from both the numerator and denominator using basic rules of algebra, resulting in simply:

Can we find the limit of this modified function via direct substitution?

Yes. If we plug in 5 for x, the function is equal to -25, and so that is our limit.

But wait, f(x)=-x² isn’t the same as our original function! At x=5, this function is defined, while the previous one wasn’t. You got rid of the discontinuity, meaning you altered the function in some way, so how can the limit be identical?

Remember, the limit as x approaches a value isn’t equal to the value at that limit itself, it is equal to what the value approaches.

By eliminating the (x-5) term, we have essentially filled in the hole at x=5, meaning that we have altered the original function so that our new function is now continuous. You are right.

But also notice that after eliminating the (x-5) term, we do not change any other information about the graph of the function, except that there is no longer a hole at x=5. If we drew the two graphs, they would be identical, except at x=5.

This is good news for us, because we are trying to determine what the function approaches at x=5, not f(5). “Filling in the hole” doesn’t change our answer for the same reason that:

This function’s limit as x→4 is 4, even though f(4) was equal to 2.

The behavior to the left and to the right of the hole approaches the value that corresponds with the hole regardless of if we filled the hole in or not.

If you graphed the original function, you could determine the limit just fine (x=5 highlighted).

And this is the graph of our modified function:

Although the values at x=5 are different, graphically, the limits are identical. Starting from the left and right, we approach the same value of 5, regardless of whether there is a hole or not.

So, we fill in the hole to avoid situations such as in the previous problem. The algebraic equivalent of what we just graphed is how we solved that problem.

We added “new information” to the graph, but the limit is identical nevertheless. We can see that when we graph both functions.

For this slightly unintuitive reason, terms like (x-5)/(x-5) are known as removable discontinuities (synonymous with “hole”) , because they can be “removed” by filling in the hole. Where there are holes in a graph, we can thus still find limits as x approaches the hole.

We have long proved that this is true when we interpret a function graphically. This is, again, the algebraic extension of that idea, and is an example of limits by factorization.

An answer where we end up with 0 over 0 using direct substitution is called the indeterminate form.

The indeterminate form is bad, and any time you reach it, you should consider another way to find the limit. In other words,

The indeterminate form does not mean that the limit does not exist, it means you need to use another way to find the limit.

The concept of the indeterminate form may seem abstract, but it is simply a function’s way of telling you that there’s something interesting going on at the point which you are trying to find the limit.

We’ll expand on this property and some more examples of limits by factorization in our next article.

This is the fifth 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.