Short on time? This page is the recap. For explanations, the flowchart and worked examples, use the full study guide. This topic is shared by Calculus AB and Calculus BC.
Notation: lim (x → a) f(x) means “the limit as x approaches a of f(x)”; x → a⁺ is from the right, x → a⁻ from the left.
Recap: the routine
- What are you given? Graph or table: estimate from each side. Formula: continue.
- Is there a split at a? Piecewise rule or |x − a|: find each one-sided limit. The limit exists only if they are equal.
- Substitute and read the result.
- If 0/0, read the form and rewrite. Then substitute again.
- Composite? Find the limit of the inside, then apply the outer function if it is continuous there.
Key relationships
| Substitution gives | Meaning | Next step |
|---|---|---|
| A number, bottom not 0 | Substitution is valid | Stop: that is the limit |
| 0 over a nonzero number | The value is 0 | Stop: the limit is 0 |
| Nonzero ÷ 0 | Unbounded near a; no finite limit | Check the sign on each side |
| 0 ÷ 0 | Indeterminate | Rewrite using the form |
| Form giving 0/0 | Tool |
|---|---|
| Polynomial over polynomial | Factor and cancel (x − a) |
| Square root and a number | Multiply by the conjugate |
| Fraction inside a fraction | Combine over a common denominator |
| Trig expression | Use an identity |
| Bounded oscillating factor, e.g. cos(1/x) | Squeeze theorem (Topic 1.8) |
Assumptions behind the method
- Substitution is valid only where the function is defined and well behaved at a (polynomials; rational functions with nonzero bottom; roots and trig inside their domains).
- The value f(a) never decides a limit.
- A table or graph gives an estimate; with a formula, use algebra for the exact value.
Mistakes to avoid
- Factoring before substituting. Substitute first; you may already be done.
- Treating 0/0 as 0 or as “does not exist”.
- Cancelling |x − a| as if it were (x − a). Split into sides first.
- Writing ∞ after checking one side only.
- Using f(a) for a piecewise function.
- Giving up on a composite because the inside is undefined at a.
Quick self-check
- Find lim (x → 4) (x² − 3x)/(x + 1). (4/5, by substitution: the bottom is 5)
- Find lim (x → −1) (x + 1)/|x + 1|. (Does not exist: 1 from the right, −1 from the left)
- Describe (x² + 3)/(x − 1) as x → 1. (4/0, so no finite limit: +∞ from the right, −∞ from the left, so the limit does not exist)
Next: practice questions.