Find the limit of x² + 3 as x approaches 2.
- Polynomials are continuous, so substitute x = 2.
- 2² + 3 = 7.
Answer: 7
Evaluating Limits Analytically Worksheets
Evaluate limits with the approach direction stated and distinguish limits from function values.
Learn the skill
Evaluating limits by substitution and factoring
A limit describes what function values approach as the input approaches a point. Direct substitution works for continuous expressions at that point. A 0/0 result is indeterminate and may require rewriting the expression.
A method you can use
- Try substitution and check whether the expression is defined.
- For a 0/0 rational form, factor and simplify for inputs near but not equal to the target.
- Evaluate the simplified expression and distinguish the limit from the function’s value at the point.
Worked examples
Find the limit of (x² − 4)/(x − 2) as x approaches 2.
- Substitution gives 0/0. Factor to (x − 2)(x + 2)/(x − 2).
- For x ≠ 2 the expression is x + 2, whose limit is 4.
Answer: 4
Find the limit of (x² − 9)/(x − 3) as x approaches 3.
- Cancel the common factor after factoring x² − 9.
- Near 3 the expression equals x + 3, which approaches 6.
Answer: 6
Try it yourself
Solve these before opening the answers. The printable sets below give you more practice.
1. Find the limit of 2x + 5 as x approaches 3.
Show answer and explanation
11
Direct substitution gives 2(3) + 5.
2. Find the limit of (x² − 1)/(x − 1) as x approaches 1.
Show answer and explanation
2
For x ≠ 1, the quotient equals x + 1, which approaches 2.
Before you begin
Manipulate algebraic expressions and understand functions, limits and derivative notation.
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
Find lim as x → 2 of (x² − 4)/(x − 2).
2
Find lim as x → 6 of (x² − 36)/(x − 6).
3
Find lim as x → 7 of (x² − 49)/(x − 7).
4
Find lim as x → 5 of (x² − 25)/(x − 5).
5
Find lim as x → 8 of (x² − 64)/(x − 8).
6
Find lim as x → 1 of (x² − 1)/(x − 1).
Answer Key
1. 42. 123. 144. 105. 166. 2