Unit One Review: Limits, Asymptotes, Continuity, and IVT (Vocabulary)

Vocabulary flashcards covering horizontal/vertical/slant asymptotes, holes and discontinuities, continuity concepts, piecewise functions, one-sided limits, IVT, and endpoint considerations.

Horizontal Asymptote

A horizontal line y = c that the graph approaches as x → ±∞. If the degree in the denominator is greater than the degree in the numerator, c = 0. If the degree in the numerator is greater, there is no horizontal asymptote (a slant/oblique asymptote may occur). If the degrees are equal, y = (leading coefficient of the numerator) / (leading coefficient of the denominator).

Slant (Oblique) Asymptote

A diagonal line y = mx + b that the graph approaches as x → ±∞ when the degree of the numerator is exactly one more than that of the denominator. The line is determined by the leading terms.

Vertical Asymptote

A vertical line x = a where f(x) grows without bound as x approaches a from the left or right. It comes from zeros of the denominator after any cancellations.

Hole

A point x = a where a factor cancels between the numerator and the denominator, making the function undefined at a. The graph has a hole at (a, freduced(a)), where freduced is the simplified function.

Removable Discontinuity

A discontinuity where the limit exists but the function value at that point does not equal the limit (or the function is undefined). It can be “removed” by redefining the function value to equal the limit.

Jump Discontinuity

A non-removable discontinuity where the left-hand limit and right-hand limit exist and are finite but are not equal, creating a jump in the graph.

Infinite Discontinuity

A non-removable discontinuity where f(x) → ±∞ as x → a, typically associated with a vertical asymptote.

Continuity

A function is continuous at a point a if lim x→a f(x) exists, f(a) exists, and lim x→a f(x) = f(a). If any of these fail, the function is not continuous at a.

Endpoint Continuity

Continuity at an endpoint of a closed interval: the function is continuous from the interior up to the endpoint, satisfying the one-sided condition at that endpoint.

Limit

The value that f(x) approaches as x approaches a (or ±∞). Used to analyze continuity and end behavior; limits can be finite or infinite.

One-Sided Limits

The limits from each side: lim x→a− f(x) (left) and lim x→a+ f(x) (right). These determine continuity and types of discontinuities.

Piecewise-Defined Function

A function defined by different expressions on different intervals. Continuity must be checked at the boundary where the definitions meet.

Continuity Test for Piecewise Functions

At a boundary point, require that the left-hand limit equals the right-hand limit and that the function value equals that common limit for continuity.

Intermediate Value Theorem

If f is continuous on [a,b] and N lies between f(a) and f(b), then there exists c ∈ (a,b) such that f(c) = N.

Hole Coordinate

The coordinates of a hole are (a, f_reduced(a)) after canceling common factors in the rational expression, where a is the x-coordinate of the hole.