Limits Involving Infinity (AP Calculus BC Unit 1 Study Notes)

Infinite limit

A limit where f(x) grows without bound (positive or negative) as x approaches a value a; it describes behavior near a, not a finite number.

(\lim_{x\to a} f(x)=\infty)

Means f(x) can be made arbitrarily large and positive by taking x sufficiently close to a (with x≠a); infinity is not a real output value.

(\lim_{x\to a} f(x)=-\infty)

Means f(x) becomes arbitrarily large in magnitude and negative as x approaches a (with x≠a).

One-sided limit

A limit taken by approaching a from only one side: from the left (x→a−) or from the right (x→a+).

Left-hand infinite limit

A statement like (\lim_{x\to a^-} f(x)=\pm\infty), describing unbounded behavior as x approaches a from values less than a.

Right-hand infinite limit

A statement like (\lim_{x\to a^+} f(x)=\pm\infty), describing unbounded behavior as x approaches a from values greater than a.

Two-sided limit does not exist (DNE) due to mismatch

If left-hand and right-hand behaviors near a are not the same (including (+\infty) vs (-\infty)), then (\lim_{x\to a} f(x)) does not exist.

Vertical asymptote

A vertical line x=a that the graph approaches as x approaches a, where at least one one-sided limit is (+\infty) or (-\infty).

Criterion for vertical asymptote at (x=a)

x=a is a vertical asymptote of f if any of (\lim{x\to a^-} f(x)=\pm\infty) or (\lim{x\to a^+} f(x)=\pm\infty) holds.

Sign analysis (near a zero denominator)

A routine for infinite limits of a quotient: determine the sign of numerator and denominator near a (from left/right), then combine signs to decide (+\infty) or (-\infty).

Prototype (\frac{1}{x-a}) behavior

As x→a−, (\frac{1}{x-a}\to -\infty); as x→a+, (\frac{1}{x-a}\to \infty) because the denominator changes sign.

Prototype (\frac{1}{(x-a)^2}) behavior

As x→a from either side, (\frac{1}{(x-a)^2}\to \infty) because the squared denominator stays positive and approaches 0.

Undefined at (a)

The function value f(a) is not defined; this alone does not determine whether a limit exists or whether there is an asymptote.

Removable discontinuity (hole)

A discontinuity where the limit exists and is finite, but the function is undefined (or defined differently) at that x-value; often caused by a canceling factor.

Factor cancellation test

To decide between a hole and a vertical asymptote in a rational function, factor and simplify; if the problematic factor cancels, the discontinuity is removable.

Limit at infinity

A limit describing end behavior as x becomes very large (x→∞) or very negative (x→−∞).

Horizontal asymptote

A horizontal line y=L that the graph approaches as x→∞ and/or x→−∞, determined by limits at infinity.

Right-end horizontal asymptote

If (\lim_{x\to \infty} f(x)=L), then y=L is a horizontal asymptote to the right.

Left-end horizontal asymptote

If (\lim_{x\to -\infty} f(x)=M), then y=M is a horizontal asymptote to the left (which may differ from the right-end one).

Horizontal asymptotes are not barriers

A graph can cross a horizontal asymptote; it only describes what the function approaches for large |x|.

Degree comparison (rational functions)

For (\frac{p(x)}{q(x)}), end behavior is determined by comparing degrees of p and q (highest powers dominate for large |x|).

Case (\deg(p)<\deg(q))

For a rational function, (\lim_{x\to \pm\infty}\frac{p(x)}{q(x)}=0), giving horizontal asymptote y=0.

Case (\deg(p)=\deg(q))

For a rational function, the limit at infinity equals the ratio of leading coefficients (\frac{a}{b}), giving horizontal asymptote y=(\frac{a}{b}).

Case (\deg(p)>\deg(q))

For a rational function, the limit at infinity is not finite (may grow without bound), so there is no horizontal asymptote.

Conjugate trick (for (\infty-\infty) forms)

An algebra method: multiply by the conjugate (e.g., (\sqrt{x^2+1}-x) times (\sqrt{x^2+1}+x)) to rewrite and simplify an indeterminate form before taking the limit.