AP Calculus AB and BC

Limit Laws and Indeterminate Forms

Only 0/0 and infinity/infinity qualify for L'Hospital's Rule; forms like (nonzero)/0 or infinity+infinity only look indeterminate. Split across sums, products, and quotients when each piece has a real limit; for quotients, the denominator's limit must be nonzero. Memorize sin(x)/x = 1 and (1-cos x)/x = 0 near x = 0.

The limit laws

Always try direct substitution first

For any polynomial, rational, power, exponential, logarithmic, or trig function, $\lim_{x \to c} f(x) = f(c)$ at every point in its domain (Topic 1.12). Plug in $c$ . If you get a real number, that is the answer. The laws below justify why substitution works and tell you what to do when it fails.

Each law assumes the individual limits exist as real numbers: let $\lim_{x \to c} f(x) = L$ and $\lim_{x \to c} g(x) = M$ (Topic 1.5).

Law	Rule	When to use it
Sum and difference	$\lim_{x \to c}\left[f(x) \pm g(x)\right] = L \pm M$	Break any expression joined by $+$ or $-$ into separate limits.
Constant multiple	$\lim_{x \to c}\left[k \cdot f(x)\right] = k \cdot L$	Pull a numerical coefficient outside the limit.
Product	$\lim_{x \to c}\left[f(x) \cdot g(x)\right] = L \cdot M$	Split a product into factors you can evaluate on their own.
Quotient	$\lim_{x \to c} \dfrac{f(x)}{g(x)} = \dfrac{L}{M}, \ M \neq 0$	Valid only when the denominator limit is nonzero. If $M = 0$ , stop: you have $0/0$ or an infinite limit.
Power	$\lim_{x \to c}\left[f(x)\right]^{n} = L^{n}$	Valid for $n$ a positive integer. Evaluate the inside first, then raise to the power.
Root	$\lim_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{L}$	Needs $L \ge 0$ when $n$ is even.
Constant	$\lim_{x \to c} k = k$	A constant function ignores $x$ .
Identity	$\lim_{x \to c} x = c$	The building block behind direct substitution.
Composite	$\lim_{x \to c} f\left(g(x)\right) = f(M)$	Valid when $f$ is continuous at $M$ : take the inside limit, then apply the outer function.

Indeterminate vs forms that only look indeterminate

When direct substitution returns one of exactly two patterns, the limit is genuinely indeterminate and L'Hospital's Rule (Topic 4.7) is on the table. Everything else that looks alarming actually settles to a definite answer.

Substitution gives	Status	What to do
$\dfrac{0}{0}$	Indeterminate	First try factoring, a conjugate, or a special trig limit. If none apply, use L'Hospital's Rule (Topic 4.7).
$\dfrac{\infty}{\infty}$	Indeterminate	Compare growth rates or apply L'Hospital's Rule. For a ratio of polynomials, the ratio of leading terms also gives the answer.

Substitution gives	Actual result	Why
$\dfrac{k}{0}, \ k \neq 0$	$+\infty$ , $-\infty$ , or DNE	A vertical asymptote. Check the sign of the denominator from each side (Topic 1.14).
$\dfrac{0}{k}, \ k \neq 0$	$0$	A vanishing numerator over a fixed nonzero number is $0$ .
$\dfrac{k}{\infty}$	$0$	A fixed number divided by something unbounded shrinks to $0$ .
$\infty + \infty$	$+\infty$	Two quantities both growing without bound add to $\infty$ .
$k \cdot \infty, \ k \neq 0$	$\pm\infty$	A nonzero constant times an unbounded quantity stays unbounded; the sign of $k$ fixes the sign.
$\infty - k$	$+\infty$	Subtracting a finite number never tames an unbounded one.

Genuinely indeterminate, but off the exam

The forms $\infty - \infty$ , $0 \cdot \infty$ , $1^{\infty}$ , $0^{0}$ , and $\infty^{0}$ are indeterminate too, but the CED excludes them from both AB and BC (Topic 4.7 exclusion statement). On the exam, L'Hospital's Rule always starts from $0/0$ or $\infty/\infty$ .

The two special trig limits

Both special limits come from the Squeeze Theorem (Topic 1.8). They turn a $0/0$ trig form into a number without L'Hospital, but only when the angle inside the trig function matches the variable in the denominator.

Limit	Value	How to spot and use it
$\lim_{x \to 0} \dfrac{\sin x}{x}$	$1$	The argument of $\sin$ must equal the denominator, and both must approach $0$ . Force a match by multiplying and dividing.
$\lim_{x \to 0} \dfrac{1 - \cos x}{x}$	$0$	Same matching rule. The value is $0$ whichever way you write the numerator, since both $1 - \cos x$ and $\cos x - 1$ vanish faster than $x$ .
$\lim_{x \to 0} \dfrac{\tan x}{x}$	$1$	Follows from $\dfrac{\sin x}{x} \cdot \dfrac{1}{\cos x}$ . Handy, but not required to memorize.

Worked example: evaluate $\lim_{x \to 0} \dfrac{\sin 5x}{x}$ . Direct substitution gives $0/0$ , and the argument $5x$ does not match the denominator $x$ . Multiply the numerator and denominator by $5$ so the denominator matches the angle:

\lim_{x \to 0} \frac{\sin 5x}{x} = \lim_{x \to 0} \frac{5 \sin 5x}{5x} = 5 \lim_{x \to 0} \frac{\sin 5x}{5x}

As $x \to 0$ , the angle $5x \to 0$ as well, so $\dfrac{\sin 5x}{5x} \to 1$ . Therefore the limit is $5 \cdot 1 = 5$ .

\lim_{x \to 0} \frac{\sin 5x}{x} = 5

Choosing the right tool

Method selection for limits is driven almost entirely by what direct substitution produces. Read off the form, then pick the matching technique.

Substitution produces	Read it as	Technique
A real number	Continuous point	Done: that number is the limit (Topic 1.5).
$0/0$ with polynomials or radicals	Removable form	Factor and cancel, or multiply by the conjugate, then re-substitute (Topic 1.6).
$0/0$ with $\sin x$ or $1 - \cos x$ near $0$	Special trig limit	Match the argument to $\dfrac{\sin x}{x}$ or $\dfrac{1 - \cos x}{x}$ (Topic 1.8).
$0/0$ or $\infty/\infty$ that will not simplify	True indeterminate	L'Hospital's Rule: differentiate the top and bottom separately (Topic 4.7).
$k/0$ with $k \neq 0$	Infinite limit	Find the sign from each side; the answer is $+\infty$ , $-\infty$ , or DNE (Topic 1.14).
Any determinate form from the table above	Not indeterminate	Read the value straight off; no special work needed.

\lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}

The L'Hospital trap

The rule above differentiates the numerator and denominator separately: it is not the quotient rule. Re-check the form after each pass, because one application can turn $0/0$ into a real number (you are done) or into a fresh indeterminate form (apply it again).