The Constraints of Limits and Applications of the One-sided Limit: An Introduction to Calculus

In our previous article, we introduced the concept of the limit, and provided a way to interpret them graphically.

When we last left off, we found that the limit sometimes doesn’t exist, like when x approaches 4 of this graph:

Let’s build on our understanding of limits, and explore why sometimes limits don’t exist.

We’ll get to a satisfying explanation of why the limit as x approaches 4 in the above example doesn’t exist. But, like reading the rules of a board game before we first start playing, it’s helpful to first list out all the ways a limit can’t exist:

Exhibit 1:

The right and left sides of a function “don’t make sense”, and approach different values (like in the example). When you evaluate the limit of a function, look at how x approaches the given value from the left and right sides of the function. (In our example, imagine a vertical line running through x=4. If x-values to the left side of the line approaches a different y-value than x-values to the right side of the line, then the limit does not exist). This may be a little confusing, and we will discuss this formally in a later article.

Exhibit 2:

The function has a vertical asymptote (eg. y=1/x). Some functions have curves that approach a vertical line, called the vertical asymptote. The function never touches the line, but the distance between the curve and the line gets closer and closer to 0 as the function heads toward positive or negative infinity. In this case, the limit as x approaches that asymptote does not exist. As always, there are some technicalities to this rule that we will discuss later.

Exhibit 3:

The domain of the function restricts where a limit exists (eg. y=sqrt(x)). The domain of y=sqrt(x) is undefined for any x-values less than 0 because you can’t take the square root of a negative number (it gives you an imaginary number), so the limit as x approaches zero of that function doesn’t exist because there is no possible number that satisfies that function which is less than zero.

There is a way to generalize these rules, and make them a little less confusing:

If the right and left sides of a function approach different y-values, the limit does not exist. This is a very informal definition, so to prove this more rigorously, let’s introduce another concept: The one-sided limit.

You already know that in this example the limit as x approaches 4 doesn’t exist:

Why doesn’t it exist?

Well, because it just wouldn’t make sense.

Let’s quantify that a little bit.

Imagine, if you will, that you draw a vertical line at x=4. The graph now looks like this:

This separates the graph into two “halves”. The left half, which we will denote as “L”, and the right half, which we will denote as “R”.

We know that the limit as x approaches that point can’t exist.

But can the limit as x approaches that point from “L” or “R” exist?

As it turns out, it can.

Think about it this way. Remember how earlier we said that the definition of a limit is whatever “the y-value of the graph seems to be at a point”?

The general concept will stay, but we will now re-define the limit:

The limit can only exist at a point if the limit as x approaches that point from “L” and “R” also exists.

That’s a little confusing. Let’s clarify what “approaches that point from L and R” means:

When the limit approaches a point from the left (L), we can interpret that as putting your pencil point anywhere on the line that is left of the imaginary vertical line x=4, and tracing that line from left to right until we reach x=4, and seeing what y-value we land on.

Similarly, when the limit approaches a point from the right (R), we can interpret that as putting your pencil point anywhere on the line that is right of the imaginary line x=4, and tracing that line from right to left until we reach x=4, and seeing what y-value we land on.

So, let’s see an example. What is the limit as x approaches 4 from the left?

Remember, we can interpret that question as “If I trace the left side of the function with my pencil, what y-value will I end up at x=4?” And tracing that line, we see that we end up at 3 from the left.

What about when x approaches 4 from the right? Tracing the right side of the function, we see that we end up at 6 from the right.

Note that it doesn’t matter that the function is technically undefined at that value when looking from the right side (notice the hole), because we’re just seeing what the function approaches, not what it really is.

Great. We have now figured out that the limit as x approaches 4 from the left is 3, and the limit as x approaches 4 from the right is 6. Mathematically, you can express that as:

Note that “left” or “right” is signified as a (-) or (+) in the exponent’s place. x→4- is interpreted as “x approaches 4 from the left”, while x→4+ is interpreted as “x approaches 4 from the right”.

Remember the limit as x→4- is not the same thing as the limit as x→-4.

What’s the point of all this? Well, here is the long awaited, more intuitive, redefinition of the limit, where c represents any real number:

The limit can only exist if both the limit from the right exists, the limit from the left exists, and they both equal each other.

We know already from common sense that the limit in the example above can’t exist. Let’s prove it using this new definition:

We know the limit as x approaches 4 from the left is 3, and the limit as x approaches 4 from the right is 6. I hope you realize that 3 is definitely not equal to 6.

Thus, the limit as the function approaches 4 cannot exist as this requirement is not fulfilled. Both “one-sided limits” exist, but they do not equal each other.

Let’s use this method to prove that the limit as x→8 exists. Remember, draw an imaginary line at x=8.

What is the limit as x approaches 8 from the left? Looks like 2 to me. What is the limit as x approaches 8 from the right? Looks like 2 to me.

The two one-sided limits exist and equal each other, so the limit must exist.

This implies that, where c represents any real number:

Thus, because the limit exists, and the one-sided limits equal each other, the limit must be equal to the value of either one-sided limit. Therefore, the limit as x→8 must be equal to 2, which is also what we got earlier via common sense.

It is important to note that these definitions are just that: definitions.

They aren’t here to constrain how you solve these types of questions. Given the graph above, of course you would answer from basic rationality that the limit as x→8 is 2, and of course you would deduce that the limit as x→4 doesn’t exist because it just “doesn’t make sense”. And that’s fine.

These definitions are here because they provide stable ground for when rationality by itself is not enough to get us to an answer, especially when discussing continuity and differentiability, which we will discuss in a later article, but that doesn’t mean basic rationality can’t be used at all.

Hopefully, you come away from this article with a better understanding of how limits are defined, and a more rigorous way to graphically determine if a limit exists or not.

This is the second 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.