Limit Properties and Rules
Essential limit laws, properties, and special limits used in calculus.
Includes L'Hôpital's Rule and 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