A limit describes the value that a function approaches as the input approaches a certain point. Written as: $\lim_{x \to c} f(x) = L$

The limit can exist even if the function is undefined at point c.

Instantaneous velocity is the limit of average velocity as the time interval approaches zero: $v(t_0) = \lim_{\Delta t \to 0} \frac{s(t_0 + \Delta t) - s(t_0)}{\Delta t}$

• Secant line: Line through two points on a curve
• Tangent line: Limit of secant lines as points approach each other
• Slope of tangent = derivative at that point

• Left-hand limit: $\lim_{x \to c^-} f(x)$
• Right-hand limit: $\lim_{x \to c^+} f(x)$
• Two-sided limit exists iff: $\lim_{x \to c^-} f(x) = \lim_{x \to c^+} f(x)$

• Look at behavior as x approaches from both sides
• Identify holes, jumps, and asymptotes
• Check for undefined points

• Continuity: graph approaches same value from both sides
• Jump discontinuity: different left/right limits
• Infinite discontinuity: vertical asymptote behavior

Use values approaching c from both sides:

As x -> 2, f(x) -> 4

If $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$:

• Sum: $\lim (f + g) = L + M$
• Difference: $\lim (f - g) = L - M$
• Product: $\lim (f \cdot g) = L \cdot M$
• Quotient: $\lim (f/g) = L/M$ (if M ≠ 0)
• Constant: $\lim k \cdot f = k \cdot L$
• Power: $\lim f^n = L^n$

For continuous functions at c: $\lim_{x \to c} f(x) = f(c)$

Works for: polynomials, rational functions (where defined), trig, exponential, and log functions at continuous points.

For $\frac{0}{0}$ indeterminate form:

Factoring example: $\lim_{x \to 2}\frac{x^2 - 4}{x - 2} = \lim_{x \to 2}\frac{(x-2)(x+2)}{x-2} = \lim_{x \to 2}(x+2) = 4$

Conjugate example: $\lim_{x \to 0}\frac{\sqrt{x+1} - 1}{x} \cdot \frac{\sqrt{x+1} + 1}{\sqrt{x+1} + 1} = \lim_{x \to 0}\frac{x}{x(\sqrt{x+1} + 1)} = \frac{1}{2}$

Find functions that bound your function from above and below, both approaching the same limit.

If $g(x) \leq f(x) \leq h(x)$ near c, and $\lim_{x \to c} g(x) = \lim_{x \to c} h(x) = L$, then $\lim_{x \to c} f(x) = L$

A function f is continuous at c if:

1. f(c) is defined
2. $\lim_{x \to c} f(x)$ exists
3. $\lim_{x \to c} f(x) = f(c)$

• Removable: Hole in graph (limit exists, function undefined or different value)
• Jump: Left and right limits exist but are different
• Infinite: Function approaches ±∞ (vertical asymptote)
• Oscillating: Function oscillates wildly (e.g., $\sin(1/x)$ near 0)

Function f is continuous at c if: $\lim_{x \to c} f(x) = f(c)$

This requires:

• f(c) exists
• Limit exists
• They are equal

• Open interval (a,b): Continuous at every point in (a,b)
• Closed interval [a,b]: Continuous on (a,b), continuous from right at a, continuous from left at b

• Polynomials: Continuous everywhere (all real numbers)
• Rational functions: Continuous everywhere except where denominator = 0
• Root functions: Continuous on domain (even roots: x ≥ 0; odd roots: all real)
• Trig functions: Continuous on their domains
• Exponential functions: Continuous everywhere
• Log functions: Continuous for x > 0

For removable discontinuities, the limit exists but doesn't equal the function value (or function is undefined).

Create a continuous function by redefining at the point of discontinuity: $g(x) = \begin{cases} f(x) & x \neq c \\ \lim_{x \to c} f(x) & x = c \end{cases}$

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

Vertical asymptote at x = c if:

• $\lim_{x \to c^-} f(x) = \pm\infty$ or
• $\lim_{x \to c^+} f(x) = \pm\infty$

Common in rational functions where denominator = 0 (and numerator ≠ 0).

$\lim_{x \to \infty} f(x) = L \text{ or } \lim_{x \to -\infty} f(x) = L$

For rational functions, compare degrees:

• Numerator degree < denominator degree: limit = 0
• Degrees equal: limit = ratio of leading coefficients
• Numerator degree > denominator degree: limit = ±∞

• Horizontal asymptote: y = L if $\lim_{x \to \pm\infty} f(x) = L$
• Slant (oblique) asymptote: When numerator degree = denominator degree + 1, find by polynomial division

If f is continuous on [a, b] and k is any value between f(a) and f(b), then there exists at least one c ∈ (a, b) such that f(c) = k.

Corollary: If f is continuous on [a, b] and f(a) - f(b) < 0 (opposite signs), then there exists c ∈ (a, b) such that f(c) = 0.

Used to prove existence of solutions to equations.