Unit 1: Limits and Continuity

Limit

A calculus concept that describes what value a function’s outputs approach as inputs get closer and closer to a point (focusing on behavior around the point, not necessarily at it).

Approach behavior

The trend in a function’s values as the input moves near a specific number (even if the function isn’t evaluated at that number).

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

Means that as (x) approaches (a), the function values (f(x)) approach (L).

Two-sided limit

A limit where (x) approaches (a) from both the left and right: (\lim_{x\to a} f(x)).

One-sided limit

A limit that considers approaching a point from only one direction (left or right).

Left-hand limit ((\lim_{x\to a^-} f(x)))

The value (f(x)) approaches as (x) approaches (a) from values less than (a).

Right-hand limit ((\lim_{x\to a^+} f(x)))

The value (f(x)) approaches as (x) approaches (a) from values greater than (a).

Two-sided limit existence condition

(\lim_{x\to a} f(x)) exists (and equals (L)) exactly when both one-sided limits exist and are equal to each other.

Function value ((f(a)))

The actual output of the function at exactly (x=a), shown by a filled point on a graph if defined.

Continuity at a point

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

Continuous on ([a,b])

Continuous at every interior point of the interval, plus right-continuous at (a) and left-continuous at (b) when endpoints are included.

Removable discontinuity

A discontinuity where the limit exists, but the function is either undefined at the point or defined with a value different from the limit (graphically often a hole).

Hole (open circle)

A missing point on a graph where the curve approaches a value, but the point is not included at that (x)-value.

Misplaced dot

A filled point that gives (f(a)), but does not match the approached value (the limit) as (x\to a).

Jump discontinuity

A discontinuity where the left-hand and right-hand limits are finite but different, so the two-sided limit does not exist.

Infinite limit

A limit where function values grow without bound near a point, written as (\lim_{x\to a} f(x)=\infty) or (-\infty) (meaning outputs become arbitrarily large in magnitude).

Vertical asymptote

A vertical line (x=a) that the function approaches with unbounded behavior (often associated with an infinite limit near (a)).

Limits at infinity

Limits that describe end behavior as inputs grow large, such as (\lim{x\to\infty} f(x)) or (\lim{x\to-\infty} f(x)).

Horizontal asymptote

A line (y=L) that describes end behavior when (\lim_{x\to\infty} f(x)=L) (the graph may cross it).

Degree comparison (rational end behavior)

For (\frac{p(x)}{q(x)}): if deg(numerator) < deg(denominator), limit at infinity is 0; if equal, it’s the ratio of leading coefficients; if numerator degree is greater, the function typically grows without bound (no horizontal asymptote).

Limit laws

Algebraic rules that allow limits of complex expressions to be computed from limits of simpler parts, assuming the relevant 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) for constant (c).

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

If (\lim f(x)=L), then (\lim (f(x))^n=L^n) for integer (n).

Direct substitution

Evaluating a limit by plugging in (x=a) when the function is continuous at (a) (no indeterminate or undefined expression occurs).

Indeterminate form

An expression from direct substitution (like (0/0)) that does not determine the limit and signals the need to simplify.

(0/0) indeterminate form

A common indeterminate form meaning both numerator and denominator approach 0; you must simplify to find the actual limit.

Factor and cancel (strategy)

A method for (0/0) rational expressions: factor numerator/denominator, cancel a shared factor (valid for (x\neq a)), then evaluate the simplified limit.

Common factor vs. term (cancelling rule)

You may cancel shared factors (after factoring), but you cannot cancel terms across addition/subtraction unless they are part of a common factor.

Rationalizing (strategy)

A technique for limits with radicals: multiply by a conjugate to remove square roots and simplify an indeterminate form.

Conjugate

For expressions like (\sqrt{x}-2), the conjugate is (\sqrt{x}+2); multiplying them uses ((a-b)(a+b)=a^2-b^2).

Trig identity strategy

When trig expressions produce (0/0), use identities to rewrite the expression into a form where the limit can be evaluated.

Pythagorean identity

A key trig identity: (\sin^2(x)+\cos^2(x)=1), often used to simplify trig limits.

Squeeze Theorem (Sandwich Theorem)

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

Bounding inequality (for squeezing)

An inequality like (g(x)\le f(x)\le h(x)) used to trap a function between two others with a known shared limit.

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

A standard calculus limit fact: (\lim_{x\to 0} \frac{\sin x}{x}=1).

Special trig limit (\lim_{x\to 0} \frac{\cos x-1}{x})

A standard calculus limit fact: (\lim_{x\to 0} \frac{\cos x-1}{x}=0).

Special trig limit (\lim_{x\to 0} \frac{\sin(ax)}{x})

A standard limit fact: (\lim_{x\to 0} \frac{\sin(ax)}{x}=a) for constant (a).

Special trig limit (\lim_{x\to 0} \frac{\sin(ax)}{\sin(bx)})

A standard limit fact: (\lim_{x\to 0} \frac{\sin(ax)}{\sin(bx)}=\frac{a}{b}) for nonzero constants (a,b).

Piecewise-defined function

A function defined by different formulas on different intervals; limits at a boundary require separate left- and right-hand analysis.

Boundary point (piecewise junction)

The (x)-value where a piecewise function changes rules; limits there are found by comparing one-sided limits from each rule.

Matching condition for continuity (choose a parameter)

To make a piecewise function continuous at (x=a), set the value from the piece containing (a) equal to the common one-sided limit at (a).

Redefining a function to remove a discontinuity

If (\lim_{x\to a} f(x)=L) but (f(a)) is missing/mismatched, define a new function with the same values for (x\neq a) and set its value at (a) to (L) to make it continuous there.

Table-based limit estimate

Estimating a limit by choosing input values closer to (a) from both sides and observing whether outputs settle toward a single number.

Graph-based limit estimate

Estimating a limit by tracing the (y)-values the graph approaches as (x) moves toward (a) from the left and right (ignoring the filled dot’s value if it differs).

Intermediate Value Theorem (IVT)

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]) such that (f(c)=N).

Sign change (bracketing a root)

When (f(a)) and (f(b)) have opposite signs for a continuous function, IVT guarantees at least one solution to (f(x)=0) in ([a,b]) (but not the exact location or how many).