Limit:

• Definition: lim(x->a) f(x) = L if for every ε > 0, there exists a δ > 0 such that |f(x) - L| < ε whenever 0 < |x - a| < δ.
• Basic limit properties: a) lim(x->a) [f(x) + g(x)] = lim(x->a) f(x) + lim(x->a) g(x) b) lim(x->a) [f(x) g(x)] = lim(x->a) f(x) lim(x->a) g(x) c) lim(x->a) [f(x) / g(x)] = lim(x->a) f(x) / lim(x->a) g(x), provided lim(x->a) g(x) ≠ 0

Continuity:

• Definition: A function f(x) is continuous at x = a if lim(x->a) f(x) = f(a).
• Continuous function properties: a) Sum, difference, product, and quotient of continuous functions are continuous. b) Composition of continuous functions is continuous.

Differentiability:

• Definition: A function f(x) is differentiable at x = a if the derivative f'(a) exists.
• Basic differentiation rules: a) (cf(x))' = cf'(x) b) (f(x) ± g(x))' = f'(x) ± g'(x) c) (f(x) g(x))' = f'(x) g(x) + f(x) * g'(x) d) (f(x) / g(x))' = (f'(x) g(x) - f(x) g'(x)) / g(x)^2, provided g(x) ≠ 0

Partial Derivatives:

• Definition: For a multivariable function f(x, y, ...), the partial derivative with respect to x is ∂f/∂x, and similarly for other variables.
• Basic partial derivative rules are similar to the basic differentiation rules.

Mean Value Theorems:

• Rolle's Theorem: If f(x) is continuous on [a, b] and differentiable on (a, b), and f(a) = f(b), then there exists a point c in (a, b) such that f'(c) = 0.
• Mean Value Theorem: If f(x) is continuous on [a, b] and differentiable on (a, b), then there exists a point c in (a, b) such that f'(c) = (f(b) - f(a)) / (b - a).

Indeterminate forms and L' Hospital's rule:

• Indeterminate forms: 0/0, ∞/∞, 0 * ∞, ∞ - ∞, 0^0, 1^∞, ∞^0
• L' Hospital's Rule: If lim(x->a) f(x) / g(x) is an indeterminate form 0/0 or ∞/∞, then lim(x->a) f(x) / g(x) = lim(x->a) f'(x) / g'(x), provided the latter limit exists.

Maxima and minima:

• Definition: A function f(x) has a local maximum (minimum) at x = a if f(a) ≥ f(x) (≤ f(x)) for all x in some neighborhood of a.
• First Derivative Test: If f'(a) changes sign from positive to negative (negative to positive) as x increases through a, then f(x) has a local maximum (minimum) at x = a.
• Second Derivative Test: If f'(a) = 0 and f''(a) < 0 (> 0), then f(x) has a local maximum (minimum) at x = a.

Taylor's theorem:

• Taylor's Theorem: If f(x) is n-times differentiable on an interval containing a, then for any x in the interval, f(x) = f(a) + f'(a)(x - a) + f''(a)(x - a)^2 / 2! + ... + f^n(a)(x - a)^n / n! + R_n(x), where R_n(x) is the remainder term.

Sequences and series:

• Definition: A sequence is an ordered list of numbers, and a series is the sum of the terms of a sequence.
• Convergence: A sequence {a_n} converges to a limit L if lim(n->∞) a_n = L. A series converges if the sequence of its partial sums converges.

Test for convergence:

• Geometric series: ∑(ar^n) converges if |r| < 1 and diverges if |r| ≥ 1.
• Integral test: If f(x) is positive, continuous, and decreasing on [1, ∞), then ∑f(n) converges if the integral ∫(f(x)dx) from 1 to ∞ converges, and diverges if the integral diverges.
• Comparison test: If 0 ≤ a_n ≤ b_n for all n, then the convergence of ∑b_n implies the convergence of ∑a_n, and the divergence of ∑a_n implies the divergence of ∑b_n.
• Ratio test: If lim(n->∞) |a_(n+1) / a_n| = L, then ∑a_n converges if L < 1 and diverges if L > 1.
• Root test: If lim(n->∞) |a_n|^(1/n) = L, then ∑a_n converges if L < 1 and diverges if L > 1.

Fourier series:

• Definition: A Fourier series is a representation of a periodic function as an infinite sum of trigonometric functions.
• Fourier series formula: f(x) = a_0 + ∑[a_n cos(nx) + b_n sin(nx)], where a_0 = (1 / T) ∫(f(x)dx) over one period, a_n = (2 / T) ∫(f(x) cos(nx)dx) over one period, and b_n = (2 / T) ∫(f(x) * sin(nx)dx) over one period.