Limits and Continuity - Vocabulary Flashcards

Key vocabulary terms with concise definitions to review limits, continuity, and the derivative.

limit

The value L that f(x) approaches as x approaches a (with x ≠ a) such that f(x) can be made arbitrarily close to L by taking x sufficiently close to a.

limit from the right (right-hand limit)

lim x→a+ f(x) = L; the limit of f(x) as x approaches a from values greater than a.

limit from the left (left-hand limit)

lim x→a− f(x) = L; the limit of f(x) as x approaches a from values less than a.

two-sided limit exists

Both the right-hand and left-hand limits exist and are equal to the same number L; then lim x→a f(x) = L.

limit does not exist (DNE) due to mismatch

If the right- and left-hand limits do not both exist or they exist but are different, then the two-sided limit does not exist.

infinite limit

A limit where f(x) grows without bound as x approaches a, i.e., lim x→a f(x) = ±∞ (the limit does not exist in the finite sense).

limit at infinity / end behavior

Limit as x→∞ or x→−∞ describing the end behavior of f(x); used to determine horizontal asymptotes.

horizontal asymptote

A horizontal line y = L such that lim x→±∞ f(x) = L; a function may have up to two horizontal asymptotes.

vertical asymptote

A vertical line x = c where f(x) → ±∞ as x→c; indicates unbounded behavior near c.

radicals limit technique (conjugate)

A method to evaluate limits involving radicals by multiplying by the conjugate to simplify the expression.

epsilon-delta definition of limit

For every ε > 0 there exists δ > 0 such that 0 < |x−a| < δ implies |f(x)−L| < ε.

epsilon (ε)

An arbitrary positive tolerance used in the precise definition of a limit.

delta (δ)

A positive distance around a such that 0 < |x−a| < δ guarantees |f(x)−L| < ε.

continuity at a point

A function f is continuous at x=c if (1) lim x→c f(x) exists, (2) f(c) exists, and (3) lim x→c f(x) = f(c).

discontinuity (types)

A point where a function is not continuous; main types are removable (hole), jump, and infinite (vertical asymptote).

removable discontinuity (hole)

A discontinuity where the limit exists but f(c) is not equal to that limit; can be 'removed' by redefining f(c).

jump discontinuity

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

infinite discontinuity

A discontinuity where f(x) → ±∞ as x → c (vertical asymptote).

continuity on an interval

A function is continuous on an open interval (a,b) if it is continuous at every point in (a,b); on a closed interval [a,b] if it is continuous on (a,b), right-continuous at a, and left-continuous at b.

polynomial continuity

Polynomials are continuous at every real number.

rational function continuity

Rational functions are continuous at every real number in their domain (where the denominator is nonzero).

absolute value continuity

The absolute value function is continuous at every real number.

nth root continuity

If n is odd, the nth root is continuous at every real; if n is even, it is continuous at every nonnegative real number.

composite continuity (composite limit theorem)

If g is continuous at c and f is continuous at g(c), then the composite f∘g is continuous at c (lim x→c f(g(x)) = f(l) where l = lim g(x)).

intermediate value theorem

If f is continuous on [a,b] and k is between f(a) and f(b), then there exists x in [a,b] with f(x) = k.

Squeeze Theorem

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

derivative

f′(x) = lim h→0 [f(x+h)−f(x)]/h; the instantaneous rate of change of f at x.

differentiability

f is differentiable at a if f′(a) exists; implies continuity at a, but continuity does not imply differentiability.

average rate of change

The slope of the secant line over [x1,x2]: (f(x2)−f(x1)) / (x2−x1).

instantaneous rate of change

The derivative value f′(x) at a specific x; the rate of change at a single point.

tangent line slope (definition via limit)

m = lim h→0 [f(x+h)−f(x)]/h; slope of the tangent line to the curve at x.

symmetric difference quotient

A numerical method to approximate f′(x): [f(x+h)−f(x−h)]/(2h) with small h.

consequence: differentiability implies continuity

If f is differentiable at a point, then f is continuous at that point; the converse is not guaranteed.

three common reasons for non-differentiability

A cusp/sharp point, a discontinuity, or a vertical tangent.

base of natural logarithms (e)

e ≈ 2.718… defined by e = lim_{n→∞} (1 + 1/n)^n; foundation of exponential and logarithmic functions.

continuous compounding

A = P e^{rt}; the limit model for compound interest as the compounding frequency increases.