Table of Contents
Understanding One-Sided Limits
In the general study of limits, we looked at how a function behaves as $x$ gets close to a number $a$ from either side. A one-sided limit focuses only on what happens as $x$ approaches $a$ from one chosen side: either from the left (smaller than $a$) or from the right (greater than $a$).
This idea is especially useful for:
- functions with jumps or breaks,
- piecewise-defined functions,
- understanding continuity at endpoints of intervals.
Here we focus on what is specific to one-sided limits: notation, meaning, examples, and how they differ from (two-sided) limits.
Notation and Basic Meaning
There are two kinds of one-sided limits:
- Left-hand limit (approach from the left, i.e., values less than $a$):
$$
\lim_{x \to a^-} f(x)
$$
This means: look at $f(x)$ for $x$ values close to $a$ but smaller than $a$. - Right-hand limit (approach from the right, i.e., values greater than $a$):
$$
\lim_{x \to a^+} f(x)
$$
This means: look at $f(x)$ for $x$ values close to $a$ but larger than $a$.
The small $-$ or $+$ on the $a$ does not mean subtract or add; it indicates from which side we are approaching $a$ on the $x$-axis.
- $a^-$: approach $a$ from values less than $a$ (left side).
- $a^+$: approach $a$ from values greater than $a$ (right side).
The function value at $a$, $f(a)$, may:
- equal the one-sided limit,
- be different from it,
- or even be undefined.
The one-sided limit is only about nearby values of $x$, not $x = a$ itself.
One-Sided Limits and Two-Sided Limits
A (two-sided) limit
$$
\lim_{x \to a} f(x)
$$
exists and equals $L$ if and only if both one-sided limits exist and are equal to each other:
- $\displaystyle \lim_{x \to a^-} f(x) = L$ and
- $\displaystyle \lim_{x \to a^+} f(x) = L$.
If the left-hand and right-hand limits are different, then the two-sided limit does not exist.
So one-sided limits are the building blocks of ordinary limits.
Graphical Interpretation
On a graph of $y = f(x)$:
- $\displaystyle \lim_{x \to a^-} f(x)$ is the $y$-value you approach as you walk along the graph from the left toward $x = a$.
- $\displaystyle \lim_{x \to a^+} f(x)$ is the $y$-value you approach as you walk along the graph from the right toward $x = a$.
Important points:
- You ignore what happens far away from $a$.
- You ignore what happens exactly at $x = a$ (including any hole or filled-in dot there).
- For a one-sided limit, you also ignore what happens on the other side of $a$.
Numerical Examples
Consider a function defined numerically by a table.
Example 1: Matching one-sided limits
Suppose $f$ behaves near $x = 2$ as follows:
- For values a bit less than 2:
$$
\begin{aligned}
x &: 1.9,\ 1.99,\ 1.999 \
f(x) &: 3.9,\ 3.99,\ 3.999
\end{aligned}
$$
These values of $f(x)$ get close to $. - For values a bit greater than 2:
$$
\begin{aligned}
x &: 2.1,\ 2.01,\ 2.001 \
f(x) &: 4.1,\ 4.01,\ 4.001
\end{aligned}
$$
These also get close to $.
Then:
- $\displaystyle \lim_{x \to 2^-} f(x) = 4$,
- $\displaystyle \lim_{x \to 2^+} f(x) = 4$,
so the two-sided limit is $4$:
$$
\lim_{x \to 2} f(x) = 4.
$$
Even if $f(2)$ were something strange (like 100, or undefined), these one-sided limits would still be $4$.
Example 2: Different one-sided limits (jump)
Suppose now:
- For $x$ slightly less than 1:
$$
\begin{aligned}
x &: 0.9,\ 0.99,\ 0.999 \
f(x) &: 2,\ 2,\ 2
\end{aligned}
$$
So $f(x)$ is always $ near $ from the left. - For $x$ slightly greater than 1:
$$
\begin{aligned}
x &: 1.1,\ 1.01,\ 1.001 \
f(x) &: 5,\ 5,\ 5
\end{aligned}
$$
So $f(x)$ is always $ near $ from the right.
Then:
- $\displaystyle \lim_{x \to 1^-} f(x) = 2$,
- $\displaystyle \lim_{x \to 1^+} f(x) = 5$,
so the two-sided limit $\displaystyle \lim_{x \to 1} f(x)$ does not exist because the one-sided limits disagree.
This is a typical jump in a graph.
Piecewise Functions and One-Sided Limits
One-sided limits are especially helpful for piecewise-defined functions, where different formulas apply on different intervals.
Example 3: Piecewise definition
Let
$$
f(x) =
\begin{cases}
x^2 - 1, & x < 1, \\
3x - 1, & x \ge 1.
\end{cases}
$$
We study what happens at $x = 1$.
- Left-hand limit at $x = 1$:
For $x$ values less than 1, we must use the first formula $f(x) = x^2 - 1$.
So
$$
\lim_{x \to 1^-} f(x)
= \lim_{x \to 1^-} (x^2 - 1)
= 1^2 - 1
= 0.
$$
- Right-hand limit at $x = 1$:
For $x$ values greater than or equal to 1, we use $f(x) = 3x - 1$.
So
$$
\lim_{x \to 1^+} f(x)
= \lim_{x \to 1^+} (3x - 1)
= 3 \cdot 1 - 1
= 2.
$$
Thus:
- $\displaystyle \lim_{x \to 1^-} f(x) = 0$,
- $\displaystyle \lim_{x \to 1^+} f(x) = 2$,
so the two-sided limit at 1 does not exist, and the graph has a jump there.
Note that $f(1)$ is given by the $x \ge 1$ formula:
$$
f(1) = 3 \cdot 1 - 1 = 2.
$$
This equals the right-hand limit but not the left-hand limit.
One-Sided Limits at Endpoints
Sometimes a function is only defined on one side of a point. For example, consider a function defined only for $x \ge 0$.
Let
$$
g(x) = \sqrt{x}, \quad \text{for } x \ge 0.
$$
We can talk about the limit as $x$ approaches $0$ from the right:
$$
\lim_{x \to 0^+} \sqrt{x} = 0,
$$
because for values like $x = 0.1, 0.01, 0.001$, the values $\sqrt{x}$ get close to $0$.
But there is no sense in talking about
$$
\lim_{x \to 0^-} \sqrt{x},
$$
because $g(x)$ is not even defined for $x < 0$. So we consider only the right-hand limit at $0$.
This is typical at endpoints of the domain: only one of the one-sided limits is meaningful.
One-Sided Limits with Infinite Behavior
One-sided limits can also involve the value going off to $+\infty$ or $-\infty$. In such cases, we say the one-sided limit is infinite.
Example 4: Vertical asymptote from one side
Consider
$$
h(x) = \frac{1}{x-2}.
$$
Near $x = 2$:
- For $x$ slightly less than 2 (e.g., $1.9, 1.99, 1.999$), $x - 2$ is a small negative number, so $\frac{1}{x-2}$ is a large negative number. Thus:
$$
\lim_{x \to 2^-} \frac{1}{x-2} = -\infty.
$$ - For $x$ slightly **greater than 2$ (e.g., $2.1, 2.01, 2.001$), $x - 2$ is a small positive number, so $\frac{1}{x-2}$ is a large positive number. Thus:
$$
\lim_{x \to 2^+} \frac{1}{x-2} = +\infty.
$$
These statements mean that $h(x)$ grows without bound in negative or positive direction as $x$ approaches 2 from each side. The line $x = 2$ is a vertical asymptote, and the behavior on each side can be different.
Simple Algebraic Examples
You can often compute one-sided limits algebraically, using the same techniques as for ordinary limits, but paying attention to which formula or which behavior applies on each side.
Example 5: Absolute value
Consider
$$
f(x) = |x|.
$$
Recall the piecewise description:
$$
|x| =
\begin{cases}
-x, & x < 0, \\
x, & x \ge 0.
\end{cases}
$$
Compute one-sided limits at $x = 0$:
- Left-hand limit:
$$
\lim_{x \to 0^-} |x|
= \lim_{x \to 0^-} (-x)
= -0 = 0.
$$ - Right-hand limit:
$$
\lim_{x \to 0^+} |x|
= \lim_{x \to 0^+} x
= 0.
$$
These are equal, so the two-sided limit exists and equals $0$:
$$
\lim_{x \to 0} |x| = 0.
$$
Example 6: A function with a hole but no jump
Let
$$
f(x) =
\begin{cases}
x^2, & x \neq 1, \\
5, & x = 1.
\end{cases}
$$
At $x = 1$, for both sides we still use the formula $x^2$ when $x$ is near 1 but not equal to 1.
- Left-hand limit:
$$
\lim_{x \to 1^-} f(x)
= \lim_{x \to 1^-} x^2
= 1.
$$ - Right-hand limit:
$$
\lim_{x \to 1^+} f(x)
= \lim_{x \to 1^+} x^2
= 1.
$$
So:
- $\displaystyle \lim_{x \to 1^-} f(x) = 1$,
- $\displaystyle \lim_{x \to 1^+} f(x) = 1$,
and therefore
$$
\lim_{x \to 1} f(x) = 1,
$$
even though $f(1) = 5$. Here there is a hole in the graph at $(1,1)$ and a point at $(1,5)$, but no jump; the one-sided limits still match.
When to Use One-Sided Limits
One-sided limits are particularly useful when:
- The function has a piecewise definition that changes at $x = a$.
- The function has a jump or step at $x = a$.
- You are analyzing behavior at an endpoint of the domain (e.g., $x \to 0^+$ when the function is only defined for $x \ge 0$).
- You see different behavior approaching $x = a$ from different sides, such as with vertical asymptotes.
They are also essential in more advanced topics such as precise definitions of continuity on closed intervals and in analyzing graphs more carefully near points of discontinuity.