Limit Properties & Fundamental Limits

Limit of a Sum/Difference

lim x->a [f(x) ± g(x)] = lim f(x) ± lim g(x)

Limit of a Constant Multiple

lim x->a [c * f(x)] = c * lim f(x)

Limit of a Product

lim x->a [f(x) * g(x)] = (lim f(x)) * (lim g(x))

Limit of a Quotient

lim x->a [f(x) / g(x)] = lim f(x) / lim g(x) (if lim g(x) ≠ 0)

Limit of a Power

lim x->a [f(x)]^n = [lim f(x)]^n

Limit of a Root

lim x->a ⁿ√f(x) = ⁿ√(lim f(x))

Limit of a Constant

lim x->a c = c

Limit of 1/x at Infinity

lim x->±∞ (1/x) = 0

Limit of 1/(x^p)

lim x->∞ (1/x^p) = 0 for p > 0

Limit of x^p at Infinity

lim x->∞ (x^p) = ∞ for p > 0

Limit of x^p at Negative Infinity

lim x->-∞ (x^p) = ∞ (even p), -∞ (odd p)

Limits of e^x

lim x->∞ e^x = ∞ and lim x->-∞ e^x = 0

Limits of ln(x)

lim x->∞ ln x = ∞ and lim x->0⁺ ln x = -∞

Limits of arctan(x)

lim x->∞ tan⁻¹ x = π/2 and lim x->-∞ tan⁻¹ x = -π/2

Fundamental Trig Limit (sin)

lim θ->0 (sin θ / θ) = 1 and lim θ->0 (θ / sin θ) = 1

Fundamental Trig Limit (cos)

lim θ->0 (1 - cos θ) / θ = 0

The Squeeze Theorem

If f(x) ≤ g(x) ≤ h(x) and lim f(x) = lim h(x) = L, then lim g(x) = L