What One-Sided Limit Notation Tells Us
A limit describes the values of a function when its input is close to a specified point. Sometimes it matters whether the input approaches from below or above. One-sided limit notation makes that direction explicit.
If \(x\) approaches \(a\) through values less than \(a\), we write \(x\to a^-\). If \(x\) approaches \(a\) through values greater than \(a\), we write \(x\to a^+\). The superscripts indicate a side of approach; they do not change the point \(a\) or specify the value of the function there.
The phrase “from the left” means that the inputs are less than \(a\), not that the function values must be less than some number. Likewise, approaching from the right means \(x>a\); it does not say that \(f(x)\) is increasing or that the function values are positive.
Precise Meaning of the Notation
Let \(f:D\to\mathbb R\), and let \(a\) be a point that can be approached by domain inputs from the side in question. The statement \(\lim_{x\to a^-}f(x)=L\) means that values of \(f(x)\) can be made as close as desired to \(L\) by taking domain inputs \(x<a\) sufficiently close to \(a\). Precisely:
Here, \(0<a-x\) is equivalent to \(x<a\), and \(a-x<\delta\) says that \(x\) is within \(\delta\) of \(a\). Together, the inequalities restrict the input to a punctured interval immediately to the left of \(a\). The condition \(x\ne a\) is built into \(0<a-x\).
For a right-hand limit, the corresponding precise meaning is:
Now \(0<x-a\) means \(x>a\), and \(x-a<\delta\) keeps \(x\) close to \(a\) on that side. In both definitions, the function’s behavior at \(x=a\) is excluded. A limit concerns nearby inputs, not necessarily the input \(a\) itself.
Worked Example: A Limit Different from the Function Value
Worked Example: A Limit Different from the Function Value
Define \(f\) by \(f(x)=\frac{x^2-4}{x-2}\) for \(x\ne2\), and \(f(2)=7\). Find the left-hand and right-hand limits at \(2\), and compare them with \(f(2)\).
For every \(x\ne2\), factor the numerator and cancel the common factor:
For the left-hand limit, \(x<2\) and approaches \(2\). The nearby function values are given by \(x+2\), so they approach \(2+2=4\):
For the right-hand limit, \(x>2\) and approaches \(2\). The same expression applies for all \(x\ne2\), so:
At the point itself, the definition gives \(f(2)=7\). Thus both one-sided limits are \(4\), while the function value is \(7\). As a check on the distinction, substituting \(x=2\) into the simplified nearby expression \(x+2\) would give \(4\), but that expression was obtained for \(x\ne2\); it does not override the separately specified value \(f(2)=7\).
When a Side Is Missing from the Domain
A one-sided limit requires domain inputs on the relevant side of \(a\) arbitrarily close to \(a\). For example, if the domain has no inputs less than \(a\), there is no left-hand approach through the domain. This is a restriction on which one-sided question can be asked, not a claim about what a graph might appear to do.
Worked Example: A Right-Hand Approach at a Domain Endpoint
Let \(D=[0,\infty)\) and \(f:D\to\mathbb R\) be \(f(x)=x\). Determine the right-hand limit at \(0\), and decide whether a left-hand limit through the domain is available.
For \(x\in D\) with \(0<x<\delta\), we have \(f(x)=x\). Given any \(\varepsilon>0\), choose \(\delta=\varepsilon\). Then \(0<x<\delta\) implies \(|f(x)-0|=x<\varepsilon\). Therefore:
There are no points \(x\in D\) with \(x<0\). Consequently, the domain supplies no inputs from which to approach \(0\) from the left. Under the domain-based definition, a left-hand limit through \(D\) is not available. The right-hand limit is confirmed by the definition, and \(f(0)=0\); that agreement does not create any left-side inputs.
How One-Sided Limits Relate to a Two-Sided Limit
A two-sided limit considers inputs on both sides of \(a\). When the domain contains points arbitrarily close to \(a\) on both sides, a finite two-sided limit exists exactly when the left-hand and right-hand limits both exist and are equal.
To prove the forward direction, assume \(\lim_{x\to a}f(x)=L\). Given \(\varepsilon>0\), the two-sided limit definition supplies \(\delta>0\) such that every domain input with \(0<|x-a|<\delta\) satisfies \(|f(x)-L|<\varepsilon\). If \(x<a\), then \(0<|x-a|<\delta\) is the same distance restriction needed on the left. If \(x>a\), it is the distance restriction needed on the right. Thus both one-sided limits equal \(L\).
For the reverse direction, assume both one-sided limits equal \(L\). Given \(\varepsilon>0\), the left-hand definition supplies \(\delta_1>0\), and the right-hand definition supplies \(\delta_2>0\), so the required inequality holds on each respective side. Choose \(\delta=\min(\delta_1,\delta_2)\). If \(x\in D\) and \(0<|x-a|<\delta\), then either \(x<a\) or \(x>a\). In the first case the left-hand condition gives \(|f(x)-L|<\varepsilon\); in the second case the right-hand condition gives the same inequality. This is precisely the two-sided limit definition, proving the result.
In particular, if the two one-sided limits are different, no two-sided limit can exist. A single value \(L\) cannot be the two-sided limit, because the theorem would require both one-sided limits to equal that same \(L\).
Worked Example: Different Limits from the Two Sides
Define \(g(x)=\frac{x}{|x|}\) for \(x\ne0\), and set \(g(0)=5\). Find the two one-sided limits at \(0\).
If \(x<0\), then \(|x|=-x\), so \(g(x)=x/(-x)=-1\). The function is constantly \(-1\) on the left of \(0\), and therefore:
If \(x>0\), then \(|x|=x\), so \(g(x)=x/x=1\). The function is constantly \(1\) on the right of \(0\), and therefore:
The results are checked directly by the expressions on each side: every negative input gives \(-1\), and every positive input gives \(1\). Since \(-1\ne1\), the two-sided limit at \(0\) does not exist. The value \(g(0)=5\) changes neither one-sided limit.
Common Notation Pitfalls
One error is to read \(x\to a^-\) as “\(x\) approaches negative \(a\).” The minus sign does not change the target point: the input still approaches \(a\), but through values less than \(a\). For instance, \(x\to3^-\) means \(x<3\) and \(x\) gets close to \(3\), not that \(x\) approaches \(-3\).
A second error is to substitute \(x=a\) when interpreting a limit. The one-sided limit definitions require \(x\ne a\); they examine nearby values. As the first example showed, the limit can be \(4\) even though the function value is \(7\). A third error is to infer that a two-sided limit exists merely because one side has a limit. Both sides must approach the same value.
Check Your Understanding
Use the definitions and distinctions in this tutorial to answer the following questions.
- In \(\lim_{x\to5^-}f(x)=L\), which side of \(5\) do the input values come from?
- Can a one-sided limit exist when \(f(a)\) is undefined? Explain what the limit concerns.
- If the left-hand and right-hand limits at \(a\) are \(2\) and \(2\), respectively, what does the theorem imply about the two-sided limit?
- If the left-hand limit is \(3\) and the right-hand limit is \(-1\), can the two-sided limit exist? Why?
- If the domain is \([4,\infty)\), are there domain inputs with which to approach \(4\) from the left?
