Fundamental Standard Limits:

• lim(x->0) sin$\frac{x}{x}$ = 1 [radians only]
• lim(x->0) tan$\frac{x}{x}$ = 1
• lim(x->0) $\frac{1-cos x}{x}$^2 = 1/2
• lim(x->0) sin^(-1)$\frac{x}{x}$ = 1
• lim(x->0) tan^(-1)$\frac{x}{x}$ = 1
• lim(x->0) $\frac{e^x - 1}{x}$ = 1
• lim(x->0) $\frac{a^x - 1}{x}$ = ln(a)
• lim(x->0) ln$\frac{1+x}{x}$ = 1
• lim(x->0) (1+x)^$\frac{1}{x}$ = e
• lim(x->a) $\frac{x^n - a^n}{(x-a)}$ = n*a^(n-1)

Derived Results:

• lim(x->0) sin$\frac{ax}{sin}$(bx) = $\frac{a}{b}$
• lim(x->0) $\frac{a^x - b^x}{x}$ = ln$\frac{a}{b}$
• lim(x->inf) (1+k/x)^(mx) = e^(mk)
• lim(x->0) $\frac{sin x - x}{x}$^3 = -1/6
• lim(x->0) $\frac{tan x - x}{x}$^3 = 1/3
• lim(x->0) $\frac{tan x - sin x}{x}$^3 = 1/2

1^infinity Formula: lim f(x)^g(x) = e^(lim g(x)*(f(x)-1)) when f->1, g->infinity

Taylor Expansions (around x=0):

• sin x = x - $x^3$/3! + $x^5$/5!
• cos x = 1 - $x^2$/2! + $x^4$/4!
• tan x = x + $x^3$/3 + 2$x^5$/15
• $e^x$ = 1 + x + $x^2$/2! + $x^3$/3!
• ln(1+x) = x - $x^2$/2 + $x^3$/3
• (1+x)^n = 1 + nx + n(n-1)$x^2$/2!

L'Hopital's Rule: lim f$\frac{x}{g}$(x) = lim f'$\frac{x}{g}$'(x) for 0/0 or inf/inf forms.

Continuity Conditions: f(a) defined, lim exists, lim = f(a).

Riemann Sum: lim$\frac{1}{n}$*sum f$\frac{r}{n}$ = integral from 0 to 1 of f(x) dx.