Definition of a limit
lim x→x0 f(x) = L if f(x) becomes arbitrarily close to L as x → x0. If not, the limit does not exist.
Left-hand and right-hand limits
Left-hand limit: lim x→x0⁻ f(x) = L⁻
Right-hand limit: lim x→x0⁺ f(x) = L⁺ The limit exists if L⁻ = L⁺ = L.
Infinite limits
If f(x) → ∞ as x → x0 (from left or right), write lim x→x0 f(x) = ∞.
If f(x) → −∞ as x → x0, write lim x→x0 f(x) = −∞
Limits at infinity
lim x→∞ f(x) = L if f(x) → L as x increases without bound.
lim x→−∞ f(x) = L if f(x) → L as x decreases without bound.
rewrite |x|
{ -x for x<0}
{ x for x>0}
Continuity at a point
f(x) is continuous at x0 if:
f(x0) is defined
lim x→x0 f(x) exists
f(x0) = lim x→x0 f(x)
Intermediate Value Theorem
If f(x) is continuous on [a, b] and f(a) ≠ f(b), then for any k between f(a) and f(b), ∃ c ∈ (a, b) such that f(c) = k.
Definition of derivative
f′(x) = lim h→0 [f(x + h) − f(x)] / h
Rolle’s Theorem
If f(x) is continuous on [a, b], differentiable on (a, b), and f(a) = f(b), then ∃ c ∈ (a, b) such that f′(c) = 0.
Mean Value Theorem
If f(x) is continuous on [a, b] and differentiable on (a, b), then ∃ c ∈ (a, b) such that f′(c) = (f(b) − f(a)) / (b − a)
L’Hôpital’s Rule (basic case)
If lim x→c f(x) = 0 and lim x→c g(x) = 0, then: lim x→c f(x)/g(x) = lim x→c f′(x)/g′(x), provided the latter exists.
Indeterminate forms handled by L’Hôpital’s Rule
0/0, ∞/∞ → apply directly
0·∞ → rewrite as ratio
∞ − ∞ → combine terms
Exponential forms (0^∞, ∞^0, 1^∞) → set y = f(x)^g(x), take ln y, reduce to 0·∞ form, then exponentiate.
steps to find hyperbolic inverse
- write in exponential form
- multiply by e^y
- Quadratic solve
- take positive root
derivatives of inverse hyperbolic identities
inverse of coshx
1. set y = cosh^-1x
2. coshy = x
3. differentiate both sides implicitly
4. rearrange and use identitiy formulas
