Unit 1: Limits and Continuity

Limit

The value a function approaches as the input gets closer to a particular number.

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

Means that as (x) gets closer to (a) (using nearby values), (f(x)) gets closer to (L).

Approaches (in limits)

Describes behavior near a point; a limit depends on values around the input, not necessarily the value at the input.

Two-sided limit

The limit of (f(x)) as (x) approaches (a) from both the left and the right, written (\lim_{x\to a} f(x)).

Left-hand limit

The limit as (x) approaches (a) using values less than (a), written (\lim_{x\to a^-} f(x)).

Right-hand limit

The limit as (x) approaches (a) using values greater than (a), written (\lim_{x\to a^+} f(x)).

One-sided limit

A limit taken from only one direction (left-hand or right-hand), often used when the context or domain restricts approach to one side.

Function value vs. limit

(f(a)) is the value at the point; (\lim_{x\to a} f(x)) is the value approached near the point. They can be different.

Two-sided limit existence condition

A two-sided limit exists exactly when both one-sided limits exist and are equal.

DNE (Does Not Exist)

A limit result indicating the function values do not approach a single finite number (e.g., jump, infinite behavior, or oscillation).

Jump discontinuity

A discontinuity where the left-hand and right-hand limits are finite but not equal.

Infinite (unbounded) behavior

When function values grow without bound toward (\infty) or (-\infty) near a point, preventing a finite limit from existing.

Oscillation (limit failure)

When values near a point keep fluctuating and do not settle to one number, so the limit does not exist.

Removable discontinuity

A “hole” where the limit exists but the function is undefined there or defined to a different value; it can be “fixed” by redefining the point to equal the limit.

Hole

A missing point on a graph (often shown with an open circle) corresponding to a removable discontinuity.

Indeterminate form

An expression form (like (0/0)) that is not an answer but a signal that algebraic rewriting is needed to evaluate a limit.

(0/0) form

A common indeterminate form showing direct substitution failed; it does not automatically mean the limit is 0 or DNE.

Direct substitution

Evaluating a limit by plugging in the approached value, valid when the function is continuous at that point (or otherwise yields a determinate form).

Limit laws

Algebraic rules that allow breaking limits of complicated expressions into limits of simpler parts (assuming the needed limits exist).

Sum/Difference Law

If (\lim f=L) and (\lim g=M), then (\lim(f\pm g)=L\pm M).

Constant Multiple Law

If (\lim f=L), then (\lim (cf)=cL).

Product Law

If (\lim f=L) and (\lim g=M), then (\lim(fg)=LM).

Quotient Law

If (\lim f=L) and (\lim g=M\neq 0), then (\lim(f/g)=L/M).

Factoring and canceling

A technique for (0/0) limits where you factor to cancel a common factor (valid for (x\neq a)) to reveal the limit and identify a removable discontinuity.

Rationalizing

A technique for limits involving radicals that produce (0/0), done by multiplying by a conjugate to simplify.

Conjugate

For (\sqrt{x}-2), the conjugate is (\sqrt{x}+2); multiplying conjugates uses ((a-b)(a+b)=a^2-b^2) to remove radicals.

Complex fraction

A fraction containing fractions (a “fraction within a fraction”); simplify by rewriting division as multiplication and clearing denominators.

Piecewise-defined function

A function defined by different formulas on different parts of the domain; one-sided limits may use different rules on each side of a point.

Squeeze Theorem

If (g(x)\le f(x)\le h(x)) near (a) and (\lim g=\lim h=L), then (\lim f=L).

Special trig limit: (\lim_{x\to 0} \frac{\sin x}{x}=1)

A foundational trigonometric limit used to evaluate many trig limits by rewriting expressions into this form.

Special trig limit: (\lim_{x\to 0} \frac{\tan x}{x}=1)

A key trigonometric limit related to the sine limit, used when expressions can be rewritten to match the pattern.

Trig scaling limit: (\lim_{x\to 0} \frac{\sin(ax)}{x}=a)

A result obtained by rewriting (\frac{\sin(ax)}{x}=\frac{\sin(ax)}{ax}\cdot a) and using (\lim_{u\to 0}\frac{\sin u}{u}=1).

Trig ratio limit: (\lim_{x\to 0} \frac{\sin(ax)}{\sin(bx)}=\frac{a}{b})

A trig limit found by rewriting in terms of (\frac{\sin(ax)}{ax}) and (\frac{\sin(bx)}{bx}) so the basic sine limit applies.

Infinite limit

A limit where (f(x)) grows without bound as (x) approaches a finite value (a) (e.g., (\lim_{x\to a} f(x)=\infty)).

Vertical asymptote

A vertical line (x=a) where the function has infinite one-sided behavior (values blow up) as (x\to a).

Limit at infinity

A limit describing end behavior as (x\to \infty) or (x\to -\infty).

Horizontal asymptote

A horizontal line (y=L) that the graph approaches as (x\to \infty) or (x\to -\infty) when the limit at infinity equals (L).

Degree (of a polynomial)

The highest power of (x) with a nonzero coefficient in a polynomial.

Leading coefficient

The coefficient of the highest-degree term of a polynomial.

Degree comparison (rational limits at infinity)

A method for (\lim_{x\to\infty} \frac{p(x)}{q(x)}) that compares degrees of numerator and denominator to determine end behavior/horizontal asymptotes.

Case (n<m) for rational functions

If numerator degree (n) is less than denominator degree (m), then (\lim_{x\to\infty}\frac{p(x)}{q(x)}=0) (horizontal asymptote (y=0)).

Case (n=m) for rational functions

If numerator and denominator have the same degree, the limit at infinity is the ratio of leading coefficients (horizontal asymptote (y=\frac{\text{lead}(p)}{\text{lead}(q)})).

Continuity at a point

A function is continuous at (x=a) when (f(a)) is defined, the limit (\lim_{x\to a}f(x)) exists, and the limit equals (f(a)).

Three conditions for continuity

(1) (f(a)) is defined; (2) (\lim{x\to a} f(x)) exists; (3) (\lim{x\to a} f(x)=f(a)).

Continuous on an interval

A function is continuous on an interval if it is continuous at every point in that interval (using one-sided behavior at endpoints when needed).

Intermediate Value Theorem (IVT)

If (f) is continuous on ([a,b]) and (C) is between (f(a)) and (f(b)), then some (c\in[a,b]) satisfies (f(c)=C).

Average rate of change

The slope between two points on a graph: (\frac{f(a+h)-f(a)}{h}); interpreted as the slope of a secant line.

Secant line

A line that intersects a curve at two points; its slope represents average rate of change over an interval.

Instantaneous rate of change

The rate of change at a single point, defined as the limit of average rates of change as the interval shrinks to 0.

Derivative (limit definition)

(f'(a)=\lim_{h\to 0}\frac{f(a+h)-f(a)}{h}), when this limit exists; it represents the instantaneous rate of change (tangent slope) at (x=a).