Chapter 2.1–2.7: Limits and Continuity — Vocabulary

A set of vocabulary flashcards covering core concepts from the notes on limits, tangent lines, velocity, continuity, and related limit laws.

Limit

The value L that f(x) approaches as x approaches a; the limit exists if f(x) can be made arbitrarily close to L for x near a (x can be not equal to a).

Limit as x approaches c (two-sided)

lim x→c f(x) = L means f(x) gets arbitrarily close to L as x tends to c from both sides.

Right-hand limit

lim x→c+ f(x) is the limit as x approaches c from values greater than c.

Left-hand limit

lim x→c− f(x) is the limit as x approaches c from values less than c.

One-sided limit

Either a left-hand or right-hand limit; the two-sided limit exists only if both exist and are equal.

Instantaneous velocity

Velocity at a specific moment; the limit of average velocity as the elapsed time interval shrinks to zero.

Average velocity

Change in position over a time interval: Δf/Δt = [f(t1) − f(t0)]/(t1 − t0).

Displacement

The change in position, Δf, also called the net change in position.

Tangent line

A line that just touches the curve at a point; its slope is the limit of the slopes of secant lines as the second point approaches the point of tangency.

Secant line

A line through two points on a curve; its slope equals the average rate of change over the interval.

Slope of tangent line

The limit of the slopes of secant lines as the distance between the two x-values defining the line shrinks to zero.

Polynomial Rule

If P is a polynomial, then lim x→x0 P(x) = P(x0).

Continuity

A function f is continuous at c if f(c) is defined, lim x→c f(x) exists, and lim x→c f(x) = f(c).

Removable discontinuity

lim x→c f(x) exists but f(c) is not defined or f(c) ≠ lim x→c f(x); can be made continuous by redefining f(c).

Jump discontinuity

lim x→c− f(x) and lim x→c+ f(x) exist but are not equal; there is a jump at x = c.

Infinite discontinuity

One-sided or two-sided limits diverge to ±∞ at x = c; the line x = c is a vertical asymptote.

Left-continuous

lim x→c− f(x) = f(c).

Right-continuous

lim x→c+ f(x) = f(c).

Indeterminate form

Forms like 0/0, ∞/∞, ∞·0, ∞ − ∞ (and 1^∞, ∞^0, 0^0) that do not determine the limit without further manipulation.

Squeeze Theorem

If f ≤ g ≤ h near c and lim x→c f(x) = lim x→c h(x) = L, then lim x→c g(x) = L.

Conjugate method

A technique to resolve indeterminate forms with square roots by multiplying by the conjugate to simplify.

Substitution method

If f is continuous at c, then lim x→c f(x) = f(c); limits can be found by evaluating at c.

Horizontal asymptote

A horizontal line y = L that the graph approaches as x → ±∞, i.e., lim x→±∞ f(x) = L.

Vertical asymptote

A vertical line x = c where lim x→c f(x) = ±∞; the function diverges there.

Limit at infinity

A limit describing the end behavior of a function as x → ∞ or x → −∞.

Sine/cosine limit (basic results)

Key limits such as lim θ→0 sin θ/θ = 1 and lim θ→0 (1 − cos θ)/θ = 0, often proven via the Squeeze Theorem.