Step 1: Try direct substitution
For polynomials and most well-behaved functions, just substitute. This is direct substitution.
Example: lim (x→3) (x² + 2x − 1) = 9 + 6 − 1 = 14
Step 2: If you get 0/0, simplify
0/0 is an indeterminate form: it doesn’t tell you the answer. Simplify algebraically.
Factoring
lim (x→5) (x² − 25)/(x − 5)
- 0/0 → factoring: (x + 5)(x − 5)/(x − 5) = x + 5
- Limit = 5 + 5 = 10
Rationalizing
lim (x→9) (√x − 3)/(x − 9)
- 0/0 → multiply by the conjugate (√x + 3):
- (x − 9)/[(x − 9)(√x + 3)] = 1/(√x + 3)
- Limit = 1/(3 + 3) = 1/6
This is rationalizing.
Common denominators
lim (x→0) [1/(x + 2) − 1/2]/x
- Combine: [2 − (x + 2)]/[2(x + 2)] ÷ x = −x/[2x(x + 2)] = −1/[2(x + 2)]
- Limit = −1/4
Step 3: Non-zero ÷ 0
If substitution gives a non-zero number over 0 (like 5/0), the function blows up near that point. Check one-sided limits; the limit is usually ±∞ or does not exist.
Limit laws
The limit laws let you split limits:
| Law | Rule |
|---|---|
| Sum | lim (f + g) = lim f + lim g |
| Product | lim (f × g) = lim f × lim g |
| Quotient | lim (f/g) = lim f / lim g (if lim g ≠ 0) |
| Constant | lim (c × f) = c × lim f |
Limits at infinity (rational functions)
For a limit at infinity, divide every term by the highest power of x in the denominator.
lim (x→∞) (3x² + 1)/(2x² − 5x)
- Divide by x²: (3 + 1/x²)/(2 − 5/x) → 3/2
| Degree comparison | Limit at ∞ |
|---|---|
| top < bottom | 0 |
| top = bottom | ratio of leading coefficients |
| top > bottom | ±∞ |
A famous limit
lim (x→0) sin x / x = 1 (with x in radians). It’s key to the derivatives of trig functions.
Worked examples
Example 1: lim (x→−2) (x² + 5x + 6)/(x + 2) → (x + 3) → 1.
Example 2: lim (x→0) (√(x + 4) − 2)/x → 1/(√(x + 4) + 2) → 1/4.
Example 3: lim (x→∞) (5x)/(x² + 1) → 0.
Common mistakes
- Saying 0/0 = 0 or 1. It’s indeterminate.
- Cancelling terms rather than factors.
- Forgetting to check whether substitution works first.
Try it yourself
- lim (x→2) (x³ − 8)/(x − 2)
- lim (x→4) (√x − 2)/(x − 4)
- lim (x→∞) (4x³ − x)/(2x³ + 7)
- lim (x→1) (x² + 3)
Answers: 1) 12 2) 1/4 3) 2 4) 4
Quick recap
- Try direct substitution first.
- For an indeterminate form 0/0, use factoring, rationalizing or common denominators.
- For a limit at infinity, compare degrees.