Rational Functions and End Behavior

Unit 1 · Polynomial and Rational Functions

What AP Precalc asks here

For a rational function $r(x) = p(x) / q(x)$ , the end behavior is determined by the relationship between the degrees of $p$ and $q$ . When the denominator outranks the numerator, $r \to 0$ . When degrees match, $r$ approaches the ratio of leading coefficients. When the numerator outranks the denominator by exactly 1, polynomial long division produces a slant asymptote.

m = \deg p

m < n

\text{HA: } y = 0

m = n

\text{HA: } y = \frac{\text{lead}(p)}{\text{lead}(q)}

m = n + 1

\text{Slant asymptote — divide and read the quotient}

m > n + 1

\text{Polynomial-like growth (no horizontal or slant asymptote)}

Caution: Degree is the highest power, not the number of terms.

r(x) = (3x + 5)/(x^2 - 1)

has numerator degree 1 and denominator degree 2 — HA is

y = 0

, not the ratio of constants.

AP problem types

Type 1

Horizontal asymptote from degree comparison

Read the degrees of numerator and denominator, then pick the matching case.

Example 1

State the horizontal asymptote of

r(x) = \dfrac{4x^3 - x}{2x^3 + 5x^2 + 1}

. (A)

y = 0

(B)

y = 2

(C)

y = -1/5

(D) No horizontal asymptote

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Type 2

Slant asymptote via polynomial long division

Divide $p(x)$ by $q(x)$ . The polynomial quotient is the slant asymptote; the remainder/divisor term tends to zero.

Example 2

Find the slant asymptote of

r(x) = \dfrac{x^2 + 3x + 1}{x - 2}

. (A)

y = x + 5

(B)

y = x + 1

(C)

y = x - 5

(D)

y = 2x + 1

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Type 3

Above-or-below approach to the asymptote

Once the asymptote is known, look at the sign of $r(x) - (\text{asymptote})$ for large $x$ to decide whether the function approaches from above or below.

Example 3

Does

r(x) = \dfrac{x^2 + 1}{x^2}

approach its horizontal asymptote from above or below as

x \to +\infty

? (A) From above (B) From below (C) The function crosses the asymptote infinitely often (D) There is no horizontal asymptote

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Rational Functions and End Behavior

What AP Precalc asks here

Horizontal asymptote (HA) cases (let m=deg⁡pm = \deg pm=degp and n=deg⁡qn = \deg qn=degq)

AP problem types

Horizontal asymptote from degree comparison

Slant asymptote via polynomial long division

Above-or-below approach to the asymptote

Horizontal asymptote (HA) cases (let $m = \deg p$ and $n = \deg q$ )