Limits Reference
Important Limit Formulas
| Limit | Result | Notes |
|---|---|---|
| lim (x→0) sin(x)/x | 1 | Fundamental trig limit |
| lim (x→0) (1-cos(x))/x | 0 | |
| lim (x→0) (1-cos(x))/x² | 1/2 | |
| lim (x→0) tan(x)/x | 1 | |
| lim (x→0) (eˣ-1)/x | 1 | Exponential limit |
| lim (x→0) (aˣ-1)/x | ln(a) | |
| lim (x→0) ln(1+x)/x | 1 | Logarithm limit |
| lim (x→∞) (1+1/x)ˣ | e | Definition of e |
| lim (x→0) (1+x)^(1/x) | e | Alternative form |
| lim (x→∞) xⁿ/eˣ | 0 | Exponential dominates polynomial |
| lim (x→∞) ln(x)/xⁿ | 0, n>0 | Polynomial dominates logarithm |
| lim (x→0⁺) x·ln(x) | 0 | 0·∞ indeterminate form |
Key Theorems
| Theorem | Statement |
|---|---|
| L'Hôpital's Rule | If lim f(x)/g(x) is 0/0 or ∞/∞, then lim f(x)/g(x) = lim f'(x)/g'(x) |
| Squeeze Theorem | If g(x) ≤ f(x) ≤ h(x) and lim g = lim h = L, then lim f = L |
| Continuity | f is continuous at a if lim(x→a) f(x) = f(a) |
Indeterminate Forms
| Form | Method |
|---|---|
| 0/0, ∞/∞ | L'Hôpital's rule |
| 0·∞ | Rewrite as 0/0 or ∞/∞ |
| ∞ - ∞ | Combine into fraction |
| 0⁰, ∞⁰, 1^∞ | Take logarithm, then apply L'Hôpital |