One-Sided Limits Worksheets
Evaluate limits with the approach direction stated and distinguish limits from function values.
Before you begin
Manipulate algebraic expressions and understand functions, limits and derivative notation.
Worked example
f(x) = 7 when x < 4, and f(x) = 11 when x ≥ 4. Find the limit as x approaches 4 from the left.
- Use the branch on the requested side of the boundary.
- A one-sided limit need not equal the limit from the other side.
Answer: 7
Check your result using the method in the example before moving to the next question.
Choose your level
These sets use the same exercise types and number ranges across level labels. Choose a fresh set for additional practice.
- Easy: Start with the preview and follow the worked example.
- Medium: Practice the same exercise types with a fresh numbered set.
- Hard: Use another set to practice explaining and checking your answers.
Questions and answer calculations checked on 2026-09-17. How we check worksheets. Free for personal and classroom use.
Preview and print an easy set
Try six questions below, or choose a full set at your level.
1
f(x) = 4 when x < 2, and f(x) = 10 when x ≥ 2. Find the limit as x approaches 2 from the left.
2
f(x) = 1 when x < 6, and f(x) = 5 when x ≥ 6. Find the limit as x approaches 6 from the right.
3
f(x) = 6 when x < 7, and f(x) = 8 when x ≥ 7. Find the limit as x approaches 7 from the left.
4
f(x) = 3 when x < 6, and f(x) = 8 when x ≥ 6. Find the limit as x approaches 6 from the right.
5
f(x) = 8 when x < 5, and f(x) = 11 when x ≥ 5. Find the limit as x approaches 5 from the left.
6
f(x) = 8 when x < 8, and f(x) = 12 when x ≥ 8. Find the limit as x approaches 8 from the right.
Answer Key
1. 42. 53. 64. 85. 86. 12