Limits Reference
Important Limit Formulas
| Limit | Result | Notes |
|---|---|---|
| lim (xโ0) sin(x)/x | 1 | Fundamental trig limit |
| lim (xโ0) (1-cos(x))/x | 0 | |
| lim (xโ0) (1-cos(x))/xยฒ | 1/2 | |
| lim (xโ0) tan(x)/x | 1 | |
| lim (xโ0) (eหฃ-1)/x | 1 | Exponential limit |
| lim (xโ0) (aหฃ-1)/x | ln(a) | |
| lim (xโ0) ln(1+x)/x | 1 | Logarithm limit |
| lim (xโโ) (1+1/x)หฃ | e | Definition of e |
| lim (xโ0) (1+x)^(1/x) | e | Alternative form |
| lim (xโโ) xโฟ/eหฃ | 0 | Exponential dominates polynomial |
| lim (xโโ) ln(x)/xโฟ | 0, n>0 | Polynomial dominates logarithm |
| lim (xโ0โบ) xยทln(x) | 0 | 0ยทโ indeterminate form |
Key Theorems
| Theorem | Statement |
|---|---|
| L'Hรดpital's Rule | If lim f(x)/g(x) is 0/0 or โ/โ, then lim f(x)/g(x) = lim f'(x)/g'(x) |
| Squeeze Theorem | If g(x) โค f(x) โค h(x) and lim g = lim h = L, then lim f = L |
| Continuity | f is continuous at a if lim(xโa) f(x) = f(a) |
Indeterminate Forms
| Form | Method |
|---|---|
| 0/0, โ/โ | L'Hรดpital's rule |
| 0ยทโ | Rewrite as 0/0 or โ/โ |
| โ - โ | Combine into fraction |
| 0โฐ, โโฐ, 1^โ | Take logarithm, then apply L'Hรดpital |