Limit Properties and Rules
Reference for essential limit laws and properties in calculus.
Covers sum, product, quotient, and power rules, L'Hopital's Rule, and the squeeze theorem.
Basic Limit Laws
lim [f(x) + g(x)] = lim f(x) + lim g(x)
lim [f(x) · g(x)] = lim f(x) · lim g(x)
lim [f(x) / g(x)] = lim f(x) / lim g(x), if lim g(x) ≠ 0
lim [c · f(x)] = c · lim f(x)
lim [f(x)]^n = [lim f(x)]^n
lim [f(x) · g(x)] = lim f(x) · lim g(x)
lim [f(x) / g(x)] = lim f(x) / lim g(x), if lim g(x) ≠ 0
lim [c · f(x)] = c · lim f(x)
lim [f(x)]^n = [lim f(x)]^n
All limits above are as x → a, and assume both individual limits exist.
Special Limits
| Limit | Value |
|---|---|
| lim (x→0) sin(x)/x | 1 |
| lim (x→0) (1-cos(x))/x | 0 |
| lim (x→0) (e^x - 1)/x | 1 |
| lim (x→0) ln(1+x)/x | 1 |
| lim (x→∞) (1 + 1/x)^x | e ≈ 2.71828 |
| lim (x→∞) (1 + a/x)^x | e^a |
| lim (x→0⁺) x·ln(x) | 0 |
| lim (x→∞) x^n / e^x | 0 (for any n) |
| lim (x→∞) ln(x) / x^n | 0 (for any n > 0) |
L'Hôpital's Rule
If lim f(x)/g(x) gives 0/0 or ∞/∞, then:
lim f(x)/g(x) = lim f'(x)/g'(x)
lim f(x)/g(x) = lim f'(x)/g'(x)
Apply repeatedly until the limit can be evaluated directly. Only valid for indeterminate forms (0/0 or ∞/∞).
Squeeze Theorem
If g(x) ≤ f(x) ≤ h(x) near a, and lim g(x) = lim h(x) = L,
then lim f(x) = L
then lim f(x) = L
Continuity
A function f(x) is continuous at x = a if all three conditions hold:
- f(a) is defined
- lim (x→a) f(x) exists
- lim (x→a) f(x) = f(a)
Example 1 — Direct Substitution
Find lim (x→3) [2x² - 1]
Direct substitution: 2(3)² - 1 = 18 - 1 = 17
Example 2 — L'Hôpital's Rule
Find lim (x→0) sin(x)/x
This is 0/0, so apply L'Hôpital's Rule:
= lim (x→0) cos(x)/1 = cos(0) = 1
Example 3 — Factoring
Find lim (x→2) (x² - 4)/(x - 2)
Factor: (x+2)(x-2)/(x-2) = x + 2
= 2 + 2 = 4
Key Notes
- Arithmetic of limits: lim[f(x)±g(x)] = lim f(x) ± lim g(x); lim[f(x)·g(x)] = lim f(x) · lim g(x); lim[f(x)/g(x)] = lim f(x)/lim g(x) (provided denominator ≠ 0). These allow limits to be evaluated term by term for sums, products, and quotients.
- L'Hôpital's rule for 0/0 and ∞/∞: If lim f(x) = 0 and lim g(x) = 0 (or both → ∞), then lim f(x)/g(x) = lim f'(x)/g'(x) — differentiate numerator and denominator separately (not the quotient rule). Apply repeatedly if still indeterminate.
- Squeeze theorem: If g(x) ≤ f(x) ≤ h(x) near x=a and lim g(x) = lim h(x) = L, then lim f(x) = L. Classic use: proving lim (sin x)/x = 1 as x→0 by squeezing between geometric inequalities.
- Continuity requires three things: f(a) is defined; lim_{x→a} f(x) exists; and lim_{x→a} f(x) = f(a). A removable discontinuity has the limit existing but not equaling f(a). Jump and infinite discontinuities have limits that don't exist (or aren't finite).
- Applications: Limit properties underpin the formal definition of the derivative (lim of difference quotient), evaluation of improper integrals, analysis of algorithm runtime as input size grows, and understanding the behavior of functions near asymptotes and discontinuities.