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.
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.