Fundamental Standard Limits:

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

Derived Results:

• lim(x->0) sin(ax)/sin(bx) = a/b
• lim(x->0) (a^x - b^x)/x = ln(a/b)
• lim(x->inf) (1+k/x)^(mx) = e^(mk)
• lim(x->0) (sin x - x)/x^3 = -1/6
• lim(x->0) (tan x - x)/x^3 = 1/3
• lim(x->0) (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 + 2x^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(x)/g(x) = lim f'(x)/g'(x) for 0/0 or inf/inf forms.

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

Riemann Sum: lim(1/n)*sum f(r/n) = integral from 0 to 1 of f(x) dx.