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Limit Definition (Epsilon-Delta)

\lim_{x \to a} f(x) = L \iff \forall \epsilon > 0, \exists \delta > 0 : 0 < |x - a| < \delta \Rightarrow |f(x) - L| < \epsilon

The formal epsilon-delta definition of a limit. For every epsilon greater than zero, there exists a delta such that f(x) is within epsilon of L whenever x is within delta of a.

Limits & Continuity · Worked examples

Left-Hand Limit

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

The limit of f(x) as x approaches a from the left (from values less than a).

Limits & Continuity · Worked examples

Right-Hand Limit

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

The limit of f(x) as x approaches a from the right (from values greater than a). A two-sided limit exists if and only if both one-sided limits exist and are equal.

Limits & Continuity · Worked examples

Squeeze Theorem

g(x) \leq f(x) \leq h(x) \text{ and } \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L \Rightarrow \lim_{x \to a} f(x) = L

If f(x) is squeezed between g(x) and h(x) near a, and g and h have the same limit L, then f also has limit L.

Limits & Continuity · Worked examples

Limit of a Sum

\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)

The limit of a sum equals the sum of the limits, provided both limits exist.

Limits & Continuity · Worked examples

Limit of a Product

\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)

The limit of a product equals the product of the limits, provided both limits exist.

Limits & Continuity · Worked examples

Limit of a Quotient

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

The limit of a quotient equals the quotient of the limits, provided the denominator limit is nonzero.

Limits & Continuity · Worked examples

Limit of a Power

\lim_{x \to a} [f(x)]^n = \left[\lim_{x \to a} f(x)\right]^n

The limit of a power equals the power of the limit, for any positive integer n.

Limits & Continuity · Worked examples

Limit of a Constant Multiple

\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)

A constant factor can be pulled out of a limit.

Limits & Continuity · Worked examples

Limit of sin(x)/x

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

One of the most important special limits in calculus. Often proved using the Squeeze Theorem with geometric arguments.

Limits & Continuity · Worked examples

Limit of (cos x - 1)/x

\lim_{x \to 0} \frac{\cos x - 1}{x} = 0

A special limit related to the derivative of cosine at x = 0.

Limits & Continuity · Worked examples

Limit Definition of e

\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^n = e

The number e (≈ 2.71828) defined as a limit. Equivalently, lim(x→0) (1+x)^(1/x) = e.

Limits & Continuity · Worked examples

L'Hopital's Rule

\text{If } \frac{0}{0} \text{ or } \frac{\pm\infty}{\pm\infty}: \lim_{x \to a} \frac{f(x)}{g(x)} = \lim_{x \to a} \frac{f'(x)}{g'(x)}

L'Hopital's Rule: When a limit gives an indeterminate form 0/0 or ∞/∞, the limit equals the ratio of the derivatives (if that limit exists).

Limits & Continuity · Worked examples

Continuity Definition

f \text{ is continuous at } a \iff \lim_{x \to a} f(x) = f(a)

A function is continuous at a point a if (1) f(a) is defined, (2) lim(x→a) f(x) exists, and (3) the limit equals f(a).

Limits & Continuity · Worked examples

Intermediate Value Theorem

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

If f is continuous on [a,b] and N is between f(a) and f(b), then there exists at least one c in (a,b) where f(c) = N. Often used to show a root exists.

Limits & Continuity · Worked examples

Extreme Value Theorem

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

If f is continuous on a closed interval [a, b], then f attains both an absolute maximum and an absolute minimum on that interval.

Limits & Continuity · Worked examples

Limits at Infinity

\lim_{x \to \infty} \frac{a_n x^n + \cdots}{b_m x^m + \cdots} = \begin{cases} 0 & n < m \\ a_n/b_m & n = m \\ \pm\infty & n > m \end{cases}

For rational functions as x→∞, compare the degrees of numerator (n) and denominator (m) to determine the limit.

Limits & Continuity · Worked examples

Infinite Limits (Vertical Asymptotes)

\lim_{x \to a} f(x) = \pm\infty \Rightarrow x = a \text{ is a vertical asymptote}

If f(x) approaches ±∞ as x approaches a, then the line x = a is a vertical asymptote of f.

Limits & Continuity · Worked examples

Limit Definition of Derivative

f'(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}

The derivative of f at x is defined as the limit of the difference quotient as h approaches 0.

Derivatives · Worked examples

Constant Rule

\frac{d}{dx}[c] = 0

The derivative of any constant is zero.

Derivatives · Worked examples

Power Rule

\frac{d}{dx}[x^n] = nx^{n-1}

Bring the exponent down as a coefficient and reduce the exponent by one. Works for any real number n.

Derivatives · Worked examples

Constant Multiple Rule

\frac{d}{dx}[c \cdot f(x)] = c \cdot f'(x)

A constant factor passes through the derivative operator.

Derivatives · Worked examples

Sum/Difference Rule

\frac{d}{dx}[f(x) \pm g(x)] = f'(x) \pm g'(x)

The derivative of a sum or difference is the sum or difference of the derivatives.

Derivatives · Worked examples

Product Rule

\frac{d}{dx}[f(x) \cdot g(x)] = f'(x)g(x) + f(x)g'(x)

The derivative of a product: derivative of the first times the second, plus the first times the derivative of the second.

Derivatives · Worked examples