One-sided limits describe what a function approaches as the input gets close to a value from only one direction. They matter because graphs can behave differently on the left and right of the same x-value, especially for piecewise functions, jumps, holes, and vertical asymptotes. A left-hand limit looks at x-values less than a, while a right-hand limit looks at x-values greater than a.
These ideas help students read graphs carefully instead of relying only on the value of the function at one point.
The notation lim x -> a- f(x) = L1 means the graph approaches L1 as x moves toward a from the left, and lim x -> a+ f(x) = L2 means it approaches L2 from the right. The ordinary two-sided limit lim x -> a f(x) exists only when the left-hand and right-hand limits both exist and are equal. The actual value f(a), if it exists, does not have to equal the limit because limits describe nearby behavior, not just the point itself.
On a graph, students can trace the curve toward x = a from each side and compare the y-values being approached.
Understanding Calculus: One-Sided Limits
A limit is built from a pattern of nearby outputs. A table can reveal that pattern just as clearly as a graph. For a chosen input of three, try inputs such as two point nine, two point nine nine, and two point nine nine nine.
Keep them all on the same side first. Then make a separate list using values just greater than three. The exact input does not need to be used in either list.
This is why a missing point may still have a precise limit. Calculator results are useful evidence, but rounding can conceal a pattern. Algebra is often needed to show the exact result.
Piecewise rules make direction especially important. Imagine a delivery charge that is five dollars for fewer than ten items and eight dollars for ten or more items. As an order size rises toward ten from below, the charge stays near five dollars.
Just above ten, it stays near eight dollars. There is no single charge being approached at the cutoff.
The rule that includes exactly ten decides the actual charge at ten, but it does not change what happens just below the cutoff. Similar jumps occur in tax brackets, parking fees, phone plans, and machine settings that switch at a threshold.
Not every one-sided limit settles near an ordinary number. Near a vertical asymptote, outputs may grow without bound. From one side they may become very large positive values.
From the other side they may become very large negative values. This describes unbounded behavior, not a function value at the asymptote. It also prevents a finite two-sided limit.
Domain boundaries create another common case. A square root rule may only exist at inputs on one side of an endpoint. At that endpoint, a one-sided limit can be meaningful even though there are no function values available from the other side.
A reliable method starts by marking the target input and separating the two directions before doing any calculation. On a graph, follow each branch toward the target without being distracted by a filled or open dot at the target itself. Check the scale on both axes, since a jump can look smaller or larger than it is.
For an equation, identify which rule applies on each side. If a fraction becomes undefined after direct substitution, factor it before deciding what the nearby behavior does.
A shared factor may cancel for nearby inputs, revealing a hole rather than a jump. This careful separation supports the later idea of continuity, where nearby behavior must agree with the assigned function value.
Key Facts
- Left-hand limit: lim x -> a- f(x) is the value f(x) approaches as x approaches a with x < a.
- Right-hand limit: lim x -> a+ f(x) is the value f(x) approaches as x approaches a with x > a.
- Two-sided limit rule: lim x -> a f(x) = L only if lim x -> a- f(x) = L and lim x -> a+ f(x) = L.
- If lim x -> a- f(x) = L1 and lim x -> a+ f(x) = L2 with L1 != L2, then lim x -> a f(x) does not exist.
- The value f(a) can be different from lim x -> a f(x), or f(a) may be undefined.
- For a piecewise function, evaluate the left-hand limit using the rule for x < a and the right-hand limit using the rule for x > a.
Vocabulary
- One-sided limit
- A limit that describes what a function approaches as x gets close to a value from only the left side or only the right side.
- Left-hand limit
- The value a function approaches as x approaches a from values less than a.
- Right-hand limit
- The value a function approaches as x approaches a from values greater than a.
- Two-sided limit
- The value a function approaches as x gets close to a from both directions when both one-sided limits agree.
- Open circle
- A graph symbol showing a point that the curve approaches but that is not included as an actual function value at that location.
Common Mistakes to Avoid
- Using f(a) as the limit automatically is wrong because a limit depends on nearby x-values, not only the function value at x = a.
- Ignoring direction symbols is wrong because x -> a- and x -> a+ can lead to different y-values on a graph or in a piecewise formula.
- Claiming the two-sided limit exists when the one-sided limits are different is wrong because both one-sided limits must agree for the two-sided limit to exist.
- Following the wrong piece of a piecewise function is wrong because the left-hand limit must use the rule for x < a and the right-hand limit must use the rule for x > a.
Practice Questions
- 1 Let f(x) = x + 2 for x < 3 and f(x) = 10 - x for x > 3. Find lim x -> 3- f(x), lim x -> 3+ f(x), and decide whether lim x -> 3 f(x) exists.
- 2 A graph approaches y = 4 as x approaches 2 from the left and approaches y = -1 as x approaches 2 from the right. What are lim x -> 2- f(x), lim x -> 2+ f(x), and lim x -> 2 f(x)?
- 3 A function has an open circle at (5, 7), a filled point at (5, 2), and the curve approaches the open circle from both sides. Explain what lim x -> 5 f(x) and f(5) are, and why they can be different.