Unit 1 AP Calculus AB - Limits (Vocabulary Flashcards)

Vocabulary-style flashcards covering key limit concepts from the Unit 1 lecture: limit basics, notation, one- and two-sided limits, indeterminate forms, techniques (factoring, conjugates), discontinuities, and limit laws.

Limit

The value f(x) approaches as x gets arbitrarily close to c from either side; it may exist even if f(c) is undefined.

Limit notation

Written as lim_{x->c} f(x) = L; reads: the limit of f(x) as x approaches c equals L.

Two-sided limit

Limit as x approaches c from both sides; exists only if the left-hand and right-hand limits are equal.

One-sided limit

Limit as x approaches c from a single side: lim{x->c^-} f(x) (left) or lim{x->c^+} f(x) (right).

Graphical interpretation of limit

The y-value the function approaches as x approaches c, regardless of whether the point is defined at c.

Direct substitution

Evaluate the limit by plugging c into f(x); valid if no division by zero or square root of a negative occurs.

Indeterminate form 0/0

Occurs when substituting yields 0/0; signals that algebraic manipulation is needed to resolve the limit.

Factoring and cancellation

Resolve 0/0 by factoring the numerator and canceling common factors, then substitute.

Conjugate method

Resolve limits with square roots by multiplying top and bottom by the conjugate to simplify.

Hole (removable discontinuity)

A missing point where the function is not defined, but the limit exists and matches the simplified expression.

Vertical asymptote

A vertical line x=c where the function grows without bound as x approaches c; can lead to infinite limits or DNE.

Infinite limit

A limit that tends to ±∞ as x approaches c; indicates unbounded growth near c.

Limit laws

Rules like lim(f+g)=lim f + lim g, lim(fg)=lim f · lim g, and lim(f/g)=lim f / lim g (when the denominator limit ≠ 0).

Left-hand limit

The limit as x approaches c from the left, lim_{x->c^-} f(x).

Right-hand limit

The limit as x approaches c from the right, lim_{x->c^+} f(x).

Example: 1/(x-5) near x=5

Two-sided limit does not exist; left-hand limit is -∞, right-hand limit is ∞ due to a vertical asymptote.