Unit 1: Limits and Continuity

Limit

A description of what value f(x) is approaching as x gets close to a particular number a (not necessarily what happens at x=a).

Approaching (in limits)

Focusing on x-values near a target value a from one or both sides, rather than requiring x to equal a.

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

Means that as x gets close to a, f(x) gets close to L; it does not automatically imply f(a)=L or that f(a) exists.

Two-sided limit

(\lim_{x\to a} f(x)): the behavior of f(x) as x approaches a from both the left and the right.

One-sided limit

A limit taken from only one side of a point, either from the left (x

Left-hand limit

(\lim_{x\to a^-} f(x)): the value f(x) approaches as x approaches a using values less than a.

Right-hand limit

(\lim_{x\to a^+} f(x)): the value f(x) approaches as x approaches a using values greater than a.

DNE (Does Not Exist)

A label used when a two-sided limit fails to exist (for example, the one-sided limits do not match).

Limit at infinity

(\lim_{x\to\infty} f(x)): describes the end behavior of f(x) as x grows without bound.

Limit at negative infinity

(\lim_{x\to-\infty} f(x)): describes the end behavior of f(x) as x decreases without bound.

End behavior

How a function behaves as x becomes very large positive or very large negative (captured by limits at (\pm\infty)).

Limit laws

Algebraic rules that allow computation of limits of sums, products, quotients, etc., from limits of simpler pieces (when those limits exist).

Sum law

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

Difference law

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

Constant multiple law

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

Product law

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

Quotient law

If (\lim f(x)=L) and (\lim g(x)=M) with (M\neq 0), then (\lim \frac{f(x)}{g(x)}=\frac{L}{M}).

Power law

For integer n, if (\lim f(x)=L), then (\lim (f(x))^n=L^n).

Root law

When defined, if (\lim f(x)=L), then (\lim \sqrt[n]{f(x)}=\sqrt[n]{L}).

Direct substitution

Evaluating a limit by plugging in x=a, which works for continuous functions like polynomials and rational functions with nonzero denominator at a.

Indeterminate form

An expression from substitution (such as 0/0) that does not determine the limit and signals that more work is needed.

(\frac{0}{0}) indeterminate form

A common indeterminate form indicating the limit could be many values; it requires algebraic simplification or another method.

Nonzero-over-zero form

A form like (\frac{\text{nonzero}}{0}), which is not indeterminate and typically indicates unbounded behavior (infinite limits) depending on side behavior.

Algebraic simplification (for limits)

Rewriting an expression (factoring, rationalizing, combining fractions) so that substitution no longer produces an indeterminate form.

Factoring

Rewriting polynomials as products to reveal common factors that may cancel in a limit problem.

Canceling a common factor

After factoring, removing a shared factor in numerator and denominator (valid for x near a but not equal to the canceled root) to evaluate a limit.

Removable discontinuity

A discontinuity where the limit exists but the function value is missing or different; it can be “removed” by redefining f(a) to equal the limit.

Hole (open circle)

A graph feature indicating f(a) is undefined or not equal to the nearby trend; the limit may still exist.

Jump discontinuity

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

Infinite (essential) discontinuity

A discontinuity where the function becomes unbounded near a point, often associated with a vertical asymptote.

Rationalizing

A technique using conjugates to eliminate radicals and simplify expressions that produce indeterminate forms.

Conjugate

For an expression a+b, the conjugate is a−b; multiplying by a conjugate can create a difference of squares.

Difference of squares

The identity ((a-b)(a+b)=a^2-b^2), often produced by conjugates to simplify limits.

Combining fractions

Using a common denominator to rewrite an expression (often complex fractions) so cancellation becomes possible.

Common denominator

A shared denominator used to combine fractions into a single rational expression for simplification.

Fundamental sine limit

(\lim_{x\to 0} \frac{\sin x}{x}=1) (true when x is measured in radians).

Radians

The angle measure required for the standard trig limits (like (\lim_{x\to0} \sin x/x=1)) to hold as stated.

Scaled-angle sine limit

(\lim_{x\to0} \frac{\sin(ax)}{ax}=1), derived from the fundamental sine limit.

Sine-over-x pattern

Rewriting expressions to create (\frac{\sin()}{}) so the fundamental sine limit can be applied.

Sine ratio limit

(\lim_{x\to0} \frac{\sin(ax)}{\sin(bx)}=\frac{a}{b}), using the fundamental sine limit.

Cosine companion limit (first form)

(\lim{x\to0} \frac{1-\cos x}{x}=0) (equivalently (\lim{x\to0} \frac{\cos x-1}{x}=0)).

(\lim_{x\to0} \frac{1-\cos x}{x^2}) limit

A key trig limit equal to (\tfrac{1}{2}), often found by rationalizing and using (\sin x/x\to1).

Squeeze Theorem

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

Boundedness of sine

The fact that (-1\le\sin(1/x)\le1), which helps apply the Squeeze Theorem to products involving (\sin(1/x)).

Oscillation

Rapid back-and-forth behavior (like (\sin(1/x)) near 0) that may prevent a limit unless another factor forces the expression to a single value.

Infinite limit

A limit statement like (\lim_{x\to a} f(x)=\infty) or (-\infty), meaning f(x) grows without bound near a (not a finite number).

Vertical asymptote

A line x=a where the function becomes unbounded as x approaches a from at least one side (often producing one-sided infinite limits).

Horizontal asymptote

A line y=L that a function approaches as x→∞ or x→−∞ when the end-behavior limit is finite; it can be crossed.

Continuity at a point

f is continuous at x=c if (1) f(c) exists, (2) (\lim{x\to c} f(x)) exists, and (3) (\lim{x\to c} f(x)=f(c)).

Intermediate Value Theorem (IVT)

If f is continuous on [a,b] and N is between f(a) and f(b), then there exists c in [a,b] such that f(c)=N (guarantees existence, not uniqueness).