Limits, Continuity, and Discontinuities

Vocabulary flashcards covering limits, asymptotes, the squeeze theorem, and continuity.

Limit (two-sided)

The value L that f(x) approaches as x → a from both sides; may or may not equal f(a).

Infinite limit

lim f(x) as x → a = ±∞: f grows without bound near a; often accompanies a vertical asymptote x = a.

Vertical asymptote

A vertical line x = a where f(x) → ±∞ as x → a from either side.

Squeeze Theorem

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

Limit of x sin(1/x) as x→0

0; shown by squeezing -|x| ≤ x sin(1/x) ≤ |x| and dividing.

Indeterminate form 0/0

Substitution yields 0/0, so the limit cannot be determined by substitution alone and requires other techniques.

Limit sin x / x as x→0

1; a fundamental limit proven via geometry and the squeeze theorem.

Removable discontinuity

A discontinuity where lim f(x) as x→a exists but f(a) ≠ that limit; can be fixed by redefining f(a) to the limit.

Jump discontinuity

A discontinuity where the left- and right-hand limits exist but are not equal; the function 'jumps'.

Infinite discontinuity

A discontinuity where a one-sided limit is infinite; the limit does not exist but the function is unbounded near a.

Continuity at a

lim f(x) as x→a exists and equals f(a).

Limit is independent of f(a)

The limit depends only on values of f(x) near a (x ≠ a); f(a) does not affect the limit.

Polynomial is continuous

Polynomials are continuous for all real numbers (continuous everywhere).

One-sided limits

Limit of f(x) as x approaches a from the left (a^−) or from the right (a^+).

Two-sided limit exists

Both one-sided limits exist and are equal; then the two-sided limit exists.

Continuity on an interval

A function is continuous at every point in an interval; e.g., polynomials are continuous everywhere.