# Limits with Zero in the Denominator

Lesson Features »

Lesson Priority: VIP Knowledge

- Recall how vertical asymptotes are formed
- Distinguish between punctures and asymptotes
- Use algebra and limit behavior to evaluate algebraic limit expressions that involve partial positives
- Determine the limit of a rational expression at the point where its denominator is zero but its numerator is not zero
- Determine the limit of a rational expression at the point where both its numerator and denominator are zero

Using new knowledge about how limits work, we will practice evaluating limits of variable expressions. Most courses focus on rational functions for this purpose, so while we will look at several situations, the focus here will be on rational functions and expressions.

Practice problems and worksheet coming soon!

## You Just Divided By Zero

There are a choice few operations that we just can't do in mathematics, and the most widely known one is divide by zero. It is singularly the most difficult impasse to explain, because unlike other "no-no"s like square roots of negative numbers or taking the log of zero, there isn't as logical or intuitive explanation for the phenomenon. Even Siri can't give you a logical explanation...One way or another, however, you have encountered the problem at some point in your math journey, and regardless of whether or not you've pondered on why division by zero is impossible, you know at least that we can't do it. Not until now, anyway.Using the magic of limits, you can finally calculate exactly what behaviors appear when zero is in the denominator. Though it took you until Calculus, you're finally going to be allowed to divide by zero.## Vertical Asymptotes

One of the few times we would have encountered division by zero in a rigorous function analysis setting is when we studied rational functions. Most commonly (but not 100% of the time), rational functions take on vertical asymptote behavior at the $x$ values which cause division by zero to occur. An example:$$y = \frac{4}{x-2}$$(1)This function has a "simple" hyperbola shape to it, and a vertical asymptote at $x=2$, where the function on the left side of the asymptote is quickly approaching $-\infty$ and the function on the right side of the asymptote is quickly approaching $+\infty$. While we may have digested the situation back then in a way that helped us understand why this behavior happens, it is only through the use of limits that we can properly qualify the situation.## Introducing the Partial Positive

When we use one-sided limits, we can conceptually consider what behavior is developing by considering substituting in a slightly smaller or slightly larger number, depending on which side we are coming from. For example, since $2^-$ means we are very very close to $2$ from the left side, we can say $2^- \approx 1.999999$ and consider approximations in the following way:$$\lim_{x \to 2^-} \, 2-x \approx 2-2^- \approx 2-1.999999$$$$\approx 0.000001 \approx \boxed{0^+}$$The idea is that when we are evaluating one-sided limits, we can quickly analyze behavior using a substitution technique that sees us literally plug in a number like $6^-$ or $3^+$ directly into the function. We can then proceed, simplify, and perform addition and subtraction to see what simplifies. The useful result of this is that any time you have a result that is virtually $0$, you will have either have the positive number $0^+$ (which is slightly larger than $0$, or in other words the right side of $0$) or the negative number $0^-$ (slightly smaller than $0$, the left side approaching $0$).## Dividing Non-Zero by Zero

Now that we have defined the partial positives of $0^-$ and $0^+$, we can jump right into the main ideas around dividing by zero. First we will consider what happens when you divide a non-zero number by zero. Later in the lesson we will examine the very different scenario for $x$ values that cause functions to be $0 \div 0$.Let's start with what we do know: when a function has a division by zero issue, and the numerator of the function is non-zero, you will definitely have a vertical asymptote. However, since there are several ways in which the function could behave in this case, just knowing this fact is not enough to understand exactly what the division by zero represents.One reason that division by zero is not something your calculator will define for you is that if you divide a non-zero number by zero, you either get $\infty$ or $-\infty$, but there's no way to tell because there are three things that the function may do near the vertical asymptote:1. Each one-side limit is $+\infty$ 2. Each one-side limit is $-\infty$ 3. One one-side limit is $+\infty$, and the other is $-\infty$.The only way to tell is through limit analysis. This is why when you type $4 \div 0$ into your calculator, it just says "error" - theres no way to know which situation $4 \div 0$ could belong to.## Limits of the Form $0 \div 0$

We turn now to the scenario where a function evaluates as $0 \div 0$ at a given $x$ value. This scenario is identically different from before because stuff like $12 \div 0$ yields some kind of answer involving infinity, but $0 \div 0$ could actually be anything: $1$, $-4$, $52$, ....Under the surface, the cause of a function evaluating to $0 \div 0$ is that both the numerator and denominator have a common factor. Because of this, our ultimate goal in this scenario will be to get some factors to cancel. This will sometimes be very straightforward, and sometimes be quite complicated.## Graphical Consequence

Earlier we quickly mentioned in Theorem 2 that a function has the same behavior before and after a zero factor cancels, and that the original function has a "hole" in it at the $x$ value that caused the $0 \div 0$ problem. Another way to say this is that:- An $x$ value that causes a $k \div 0$ type issue will cause the graph to have a vertical asymptote at that point.
- An $x$ value that causes a $0 \div 0$ type issue will cause the graph to have a "hole" or "puncture" at that point.

## Put It To The Test

Let's take a look at the typical ways in which this lesson shows up on quizzes and tests.It is often helpful to plug in the limit value right away, so you can see which kind of situation you're dealing with. As we work the problems that follow, note how we begin each problem by doing this.The instructions for all the practice problems will be the same for this lesson: Evaluate the following limit, if it exists.- Understand what happens when a non-zero number is divided by zero, and why one-sided limits are needed
- Know how to quickly tell whether a zero denominator limit problem is of the form $k/0$ or $0/0$, and how to proceed in each case
- Remember when a function will feature a vertical asymptote versus a hole (once again $k/0$ versus $0/0$)
- Know the general practices for evaluating limits of the form $0/0$, trying to cancel the problematic factors
- Be familiar with the various types of $0/0$ problems we see, since certain types are always operated on the same way

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