When crossing the street, you look to the left and right to ensure there are no problems before walking across. When finding the limits of the function, you do the same thing, look to the left and right!
One-Sided Limits in Calculus
Generally, we call these the limit from the left or the limit from the right because you are specifically looking to the left or right of a specific point.
Definition of One-sided Limits
How can we formally define "one-sided limits" in calculus? Let's have a look!
We say that \(L\) is the left limit of a function \(f(x)\) at \(a\) if we can get \(f(x)\) as close to \(L\) as we want by taking \(x\) on the left side of \(a\), and close to \(a\) but not equal to \(a\). It is written
\[lim_{x \rightarrow a^-}f(x)\]
.
This is also called the limit from the left of a function. You can also look at the limit from the right of a function.
We say that \(L\) is the right limit of a function \(f(x)\) at \(a\) if we can get \(f(x)\) as close to \(L\) as we want by taking \(x\) on the right side of \(a\), and close to \(a\) but not equal to \(a\). It is written
\[ lim_{x \rightarrow a^-} f(x) \]
The nice thing is that if the limit of the function exists, then both the limit from the left and right exist and are the same.
Each of the results below follow from just the definition of the limit, the left limit, and the right limit. They are an immediate consequence of the definitions and so don't require a fancy proof.
1. Suppose that
\[lim_{x \rightarrow a} f(x)=L\]
where \(L\) is a real number. Then
\[lim_{x \rightarrow a^+} f(x)=L\]
and
\[lim_{x \rightarrow a^-} f(x)=L\]
2. Similarly, if
\[lim_{x \rightarrow a^+} f(x)=L= lim_{x \rightarrow a^-} f(x)\]
then
\[lim_{x \rightarrow a} f(x)=L\].
Note this gives you a handy way to tell if the limit doesn't exist, just by using the contrapositive of part 2.
3. If
\[lim_{x \rightarrow a^+} f(x) \neq lim_{x \rightarrow a^-} f(x)\]
then
\[lim_{x \rightarrow a} f(x)\]
does not exist.
You can read \(x \rightarrow a^+\) as "\(x\) approaching \(a\) from the right", and \(x \rightarrow a^-\) as "\(x\) approaching \(a\) from the left".
Finding One-Sided Limits
So how do you figure out what a function's left or right limit is? You can determine one-sided limits by looking at:
The graph of a function, OR
A table of function values
So let's look at a specific example.
Using the function
\[ f(x) \Biggl\{ \begin{matrix}1 x< 2 \\3 x \geq 2\end{matrix} \]
find
\[lim_{x \rightarrow 2^-} f(x)\]
and
\[lim_{x \rightarrow 2^+} f(x)\]
What does this tell you about
\[lim_{x \rightarrow 2} f(x)?\]
Answer:
First, let's look at the limit from the left. In the graph below, you can see the function, and also a table of function values that are getting closer to \(x=2\) from the left-hand side, and the points in the table plotted on the graph.
| \(x\) | \(f(x)\) |
| 1 | 1 |
| 1.5 | 1 |
| 1.75 | 1 |
| 1.9 | 1 |
| 1.95 | 1 |
| 1.99 | 1 |
Table 1. Data for the limit example.
Fig. 1. Limit from the left using a graph and table.
As you can see from the graph above, as \(x \rightarrow 2^-\), all of the function values are equal to \(1\). Therefore
\[lim_{x \rightarrow 2^-} f(x)=1\]
Now instead, let's look at the limit from the right. In the graph below, you can see the function, a table of function values that are getting closer to \(x=2\) from the right-hand side, and the points in the table plotted on the graph.
| \(x\) | \(f(x)\) |
| 3 | 3 |
| 2.5 | 3 |
| 2.25 | 3 |
| 2.1 | 3 |
| 2.05 | 3 |
| 2.01 | 3 |
Table 2. Data for the example function.
Fig. 2. Limit of a function from the right using a graph and table.
As you can see from the graph above, as \(x \rightarrow 2^+\), all of the function values are equal to \(3\). Therefore
\[lim_{x \rightarrow 2^-} f(x)=3\]
Finally, since you know that
\[lim_{x \rightarrow 2^-} f(x) \neq lim_{x \rightarrow 2^+} f(x)\]
you also know that
\[lim_{x \rightarrow 2} f(x)\]
does not exist.
Examples of One-Sided Limits
Let's look at more examples of determining one-sided limits.
Consider the function
\[f(x)= \dfrac{|x|}{x}\]
Find the limits from the left and right of \(x=0\).
Answer:
Rather than thinking about this function as one with an absolute value, it can help to think about possible values for \(x\). Let's look at the \(3\) possible cases here:
- When \(x=0\), this function is not defined.
- When \(x\) is negative, \(f(x)=-1\).
- When \(x\) is positive, \(f(x)=1\).
So you can instead think of this as the piecewise-defined function:
\[ f(x) \Biggl\{ \begin{matrix}-1, x< 0 \\1, x > 0\end{matrix} \]
This is very similar to the previous example. In fact
\[lim_{x \rightarrow 0^-+ f(x)=1\]
and
\[ lim_{x \rightarrow 0^-} f(x)=-1\]
For the function in the picture below, determine the following (if it exists):
1. \(f(1)\), \(f(3)\), and \(f(4)\).
2. the limit from the left at \(x=-3\), \(x=1\), \(x=3\), and \(x=4\).
3. the limit from the right at \(x=-3\), \(x=1\), \(x=3\), and \(x=4\).
4. the limit at\(x=-3\), \(x=1\), \(x=3\), and \(x=4\).
Fig. 3. Finding One-Sided limits from a graph.
Answer:
1. This part is just looking for the function values at these points. So looking at the graph, \(f(1)=2\), \(f(3)=4\) and \(f(4)=1\).
2. Remember that when you are finding the limit from the left, you only look at points on the graph that are to the left of the point you care about. So using the graph,
\[lim_{x \rightarrow -3^-} f(x)=3\]
\[lim_{x \rightarrow 1^-} f(x)=-5\]
[lim_{x \rightarrow 3^-} f(x)=4\]
[lim_{x \rightarrow 4^-} f(x)=4\]
3. When you are finding the limit from the right, you only look at points on the graph that are to the right of the point you care about. So using the graph:
\[lim_{x \rightarrow -3^+} f(x)=3\]
\[lim_{x \rightarrow 1^+} f(x)=2\]
\[lim_{x \rightarrow 3^+} f(x)=3\]
\[lim_{x \rightarrow 4^+} f(x)=4\]
4. The limit will exist only in cases where the limit from the left and the limit from the right are the same. Otherwise, the limit doesn't exist. Looking at the information in parts 2 and 3 above, that means that the limit exists at \(x=-3\) and at \(x=4\). You can also say that
\[lim_{x \rightarrow -3} f(x)=3\]
and
\[lim_{x \rightarrow 4} f(x)=4\]
Notice that the fact that the limit exists is independent of the actual function value at the point, or even if the function is defined there.
In addition, the limit does not exist at \(x=1\) and \(x=3\).
One-Sided Limits and Vertical Asymptotes
One question that still needs to be answered. How do we evaluate the left and right limits of a function at a vertical asymptote? The process for finding the limits from the left and right when there is a vertical asymptote is exactly the same as at any other point. Let's look at an example.
Consider the function
\[f(x)=\dfrac{1}{x}\]
Find
\[lim_{x \rightarrow 0^+} f(x)\]
and
\[lim_{x \rightarrow 0^-} f(x)\]
Answer:
First, let's think about the limit from the left. Look at the graph and table below.
| \(x\) | \(f(x)\) |
| -0.5 | -2 |
| -0.45 | -2.22 |
| -0.4 | -2.2 |
| -0.35 | -2.86 |
| -0.3 | -3.33 |
| -0.25 | -4 |
| -0.2 | -5 |
| -0.15 | -6.67 |
| -0.1 | -10 |
| -0.05 | -20 |
Table 3. Data for the limit example.
Fig. 4. Limit from the left at a vertical asymptote.
As you can see from both the graph and table, as you take \(x\) values that get closer and closer to \(x=0\) from the left, the function values become further and further away from the \(x\) axis, and are all negative. So you would say that in fact there is no number that is the limit from the left. When this happens you can say that "the limit from the left diverges to negative infinity", and write it as
\[lim_{x \rightarrow o^-} \dfrac{1}{x}=-\infty\].
This may seem odd given that limits usually have to be numbers, but the notation is just saying that the function values to the left at zero, but close to zero, can be as large a negative value as you want them to be.
When we say the limit equals \(+ \infty\) or \(-\infty\), it is just another way of saying the limit does not exist, just being a bit more specific!
Now let's think about the limit from the right. Look at a graph and table below.
| \(x\) | \(f(x)\) |
| 0.5 | 2 |
| 0.45 | 2.22 |
| 0.4 | 2.5 |
| 0.35 | 2.86 |
| 0.3 | 3.33 |
| 0.25 | 4 |
| 0.2 | 5 |
| 0.15 | 6.67 |
| 0.1 | 10 |
| 0.05 | 20 |
Table 4. Data points for the side-limit example.
Fig. 5. The limit from the right of a vertical asymptote.
As you can see from both the graph and table, as you take \(x\) values that get closer and closer to \(x=0\) from the right, the function values become further and further away from the \(x\) axis, and are all positive. So you would say that in fact there is no number that is the limit from the right. When this happens you can say that "the limit from the right diverges to infinity", and write it as
\[lim_{x \rightarrow 0^+} \dfrac{1}{x}=\infty\].
This may seem odd given that limits usually have to be numbers, but the notation is just saying that the function values to the left at zero, but close to zero, can be as large a positive number as you want them to be.
If instead of vertical asymptotes you are interested in the limit as \(x \rightarrow \pm \infty\), also known as limits at infinity, see Infinite Limits
There may be cases where the limit from one side exists, but does not exist from the other side. We see this in the example below.
For the function in the graph below, find
\(lim_{x \rightarrow 0^+}\) and \(lim_{x \rightarrow 0^-}\).
Fig. 6. Limit from one side exists but limit from other side doesn't.
From the picture above, we see that to the left of \(x=0\) the function values get closer and closer to \(3\) as \(x \rightarrow 0^-\). That means:
\[lim_{x \rightarrow 0^-} f(x)=3\]
However, if you look at values to the right of \(x=0\), the function values get larger and larger as \(x \rightarrow 0^+\). That means:
\[lim_{x \rightarrow 0^+} f(x)=\infty\]
Looking at the examples above, you can draw some helpful conclusions:
1. If
\(lim_{x \rightarrow a^+} f(x)= \pm \infty\) or if \(lim_{x \rightarrow a^-} f(x)= \pm \infty\)
then the function has a vertical asymptote at \(x=a\).
2. If the function has a vertical asymptote at \(x=a\) then either
\(lim_{x \rightarrow a^+} f(x)= \pm \infty\) or \(lim_{x \rightarrow a^-} f(x)= \pm \infty\)
One-Sided Limits - Key takeaways
- We say that \(L\) is the left limit of a function \(f(x)\) at \(a\) if we can get \(f(x)\) as close to \(L\) as we want by taking \(x\) on the left side of \(a\), and close to \(a\) but not equal to \(a\). It is written
\[lim_{x \rightarrow a^-} f(x)\]
.
We say that \(L\) is the right limit of a function \(f(x)\) at \(a\) if we can get as close to \(L\) as we want by taking \(x\) on the right side of \(a\), and close to \(a\) but not equal to \(a\). It is written
\[lim_{x \rightarrow a^+} f(x)\]
Suppose that \(lim_{x \rightarrow a} f(x)=L\) where \(L\) is a real number. Then:\(lim_{x \rightarrow a^+} f(x)=L\) and \(lim_{x \rightarrow a^-} f(x)\)
If \(lim_{x \rightarrow a^+} f(x)=L=lim_{x \rightarrow a^-} f(x)\) then:
\[lim_{x \rightarrow a} f(x)=L\].
If \(lim_{x \rightarrow a^+} f(x) \neq lim_{x \rightarrow a^-} f(x)\) then
\(lim_{x \rightarrow a} f(x)\) does not exist.
If \(lim_{x \rightarrow a^+} f(x) = \pm \infty\) or if \( lim_{x \rightarrow a^-} f(x)= \pm \infty\)
then the function has a vertical asymptote at \(x=a\).
If the function has a vertical asymptote at \(x=a\) then either:
\(lim_{x \rightarrow a^+} f(x) = \pm \infty\) os \( lim_{x \rightarrow a^-} f(x)= \pm \infty\).
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