Limits and Continuity Equations

Determining Continuty

1. f(a) is exists

2. lim x→a [(f(x)] exists → proven by left and right limits being equal ← if this is not true then in a non-removable discontinuity, either a jump (most common), infinite (VA), or oscillating discontinuity

3. f(a) = lim x→ a [f(x)] ← if only this one is not true then is removable discontinuity (hole or displaced point)

Identities

$\lim_{x\to0}\frac{\left(\sin x\right)}{x}=1$ and $\lim_{x\to0}\frac{\left(1-\cos x\right)}{x}=0$

Limits at Infinity

multiply by 1/x^n, n is the highest power

Intermediate Value Theorem

if smth is a polynomial function that is continuous on the interval [a, b], then for any value L between f(a) and f(b), there exists at least one c in (a, b) such that f(c) = L. Useful for determining if theres a zero