Limits & Continuity Formulas

Definition of a Limit

\lim_{x \to a} f(x) = L

Definition of a Limit describes what value a function approaches as its input approaches a specific point—essential for all of calculus.

Limit Laws

\lim (f \pm g) = \lim f \pm \lim g

The Sum Law shows that a limit of a sum equals the sum of the limits, provided both limits exist and are finite numbers.

One-Sided Limits

\lim_{x \to a^-} f(x) = \lim_{x \to a^+} f(x) \iff \lim_{x \to a} f(x) \text{ exists}

One-sided limits let you examine how a function behaves as you approach a point from just the left or right, determining whether a two-sided limit exists.

Limits at Infinity

\lim_{x \to \infty} \frac{a_n x^n}{b_m x^m}

Find what a rational function approaches as x grows indefinitely by comparing the degrees and leading coefficients of the numerator and denominator.

Squeeze Theorem

g(x) \le f(x) \le h(x), ; \lim g = \lim h = L \implies \lim f = L

Squeeze Theorem finds a limit by trapping a function between two simpler functions with the same limit; use it when direct methods fail.

Special Trigonometric Limit

\lim_{x \to 0} \frac{\sin x}{x} = 1

The Special Trigonometric Limit tells you that sine and its argument are equivalent near zero, letting you evaluate otherwise indeterminate forms in limits.

L'Hopital's Rule

\lim \frac{f(x)}{g(x)} = \lim \frac{f'(x)}{g'(x)} \quad \text{for } \tfrac{0}{0} \text{ or } \tfrac{\infty}{\infty}

L'Hopital's Rule efficiently solves limits that reduce to 0/0 or ∞/∞ forms by taking derivatives of the numerator and denominator separately.

Continuity

\lim_{x \to a} f(x) = f(a)

Continuity tells you when a function has no jumps or breaks, so you can safely evaluate limits and apply theorems about function behavior.

Removable Discontinuity

\lim_{x \to a} f(x) \text{ exists but } \neq f(a)

A removable discontinuity is a hole in a graph where the limit exists but the function value differs; it can be fixed by redefining that point.

Intermediate Value Theorem

f \text{ continuous on } [a,b], ; f(a) < N < f(b) \implies \exists c : f(c) = N

Intermediate Value Theorem states that a continuous function on a closed interval must cross any value between its endpoints.

Extreme Value Theorem

f \text{ continuous on } [a,b] \implies f \text{ attains a max and a min}

If a function is continuous on a closed interval, it must attain both its maximum and minimum values somewhere on that interval.

Indeterminate Forms

\tfrac{0}{0}, ; \tfrac{\infty}{\infty}, ; 0 \cdot \infty, ; \infty - \infty, ; 1^{\infty}

Indeterminate forms are expressions like 0/0 and ∞/∞ that arise when evaluating limits, signaling that direct substitution doesn't work.

Limit of a Rational Function

\lim_{x \to a} \frac{P(x)}{Q(x)} = \frac{P(a)}{Q(a)}, \quad Q(a) \neq 0

Find the limit of a rational function at a point by substituting the point directly, provided the denominator is nonzero.

Limits by Factoring

\lim_{x \to a} \frac{(x-a)g(x)}{(x-a)h(x)} = \frac{g(a)}{h(a)}

Limits by Factoring removes common factors from numerator and denominator to resolve 0/0 indeterminate forms and evaluate limits algebraically.

Limits by Rationalizing

\frac{\sqrt{x} - \sqrt{a}}{x - a} \cdot \frac{\sqrt{x} + \sqrt{a}}{\sqrt{x} + \sqrt{a}}

Multiply by the conjugate of the numerator to eliminate radicals in limits, resolving 0/0 indeterminate forms so the limit can be evaluated.

Epsilon-Delta Definition

\forall \varepsilon > 0 ; \exists \delta > 0 : 0 < |x - a| < \delta \implies |f(x) - L| < \varepsilon

Epsilon-delta provides the rigorous mathematical proof that a limit exists by relating how close outputs must be to how close inputs need to stay.

Infinite Limits

\lim_{x \to a} f(x) = \pm\infty

Infinite Limits describe when a function grows unboundedly as the input approaches a specific value, letting you identify and analyze vertical asymptotes.

Asymptotes from Limits

\lim_{x \to \infty} f(x) = L \implies y = L \text{ is a horizontal asymptote}

Use limits at infinity to determine where a function is heading as x grows without bound, then identify that destination as a horizontal asymptote equation.

Limit of a Composite Function

\lim_{x \to a} f\big(g(x)\big) = f\Big(\lim_{x \to a} g(x)\Big)

Evaluate limits of composite functions by moving the limit inside to the inner function, provided the outer function is continuous at the limiting value.

Limit Definition of e

\lim_{x \to 0} (1 + x)^{1/x} = e

The limit definition of e shows how the mathematical constant e arises from the growth of compound interest and exponential functions.

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