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 $x\to a$ 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 o 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 $x \to a^-$, ($\frac{1}{x-a} \to -\infty$); as $x \to a^+$, ($\frac{1}{x-a} \to \infty$) because the denominator changes sign.

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

As $x \to a$ 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 $x \to \infty$ or $x \to -\infty$.

Horizontal asymptote

A horizontal line $y=L$ that the graph approaches as $x \to \infty$ and/or $x \to -\infty$, 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.

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 o \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.