Polynomial and Rational Functions

Rational Function Asymptotes

Find the vertical and horizontal asymptotes of f(x) = (x+1)/(x²-4) by analyzing the denominator zeros and comparing polynomial degrees.

Vertical asymptotes (VA): occur at zeros of q(x) that are not also zeros of p(x). Set q(x) = 0 and solve. Horizontal asymptotes (HA) depend on degrees: deg(p) < deg(q): HA is y = 0. deg(p) = deg(q): HA is y = (leading coeff of p)/(leading coeff of q). deg(p) > deg(q): no horizontal asymptote (oblique instead). Example: f(x) = (x + 1)/(x² - 4). Denominator: x² - 4 = (x - 2)(x + 2) → zeros at x = 2 and x = -2. VA: x = 2, x = -2. Degree comparison: 1 < 2 → HA y = 0.

Example

Find vertical asymptotes from denominator zeros and horizontal asymptote from degrees.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

f(x)=x+1x2−4f(x)=\frac{x+ 1 }{ x^{2} - 4 }

Step 2 — Factor denominator

Factor the denominator using 2 and negative 2.

x2−4=(x−2)(x+2)x^{2} - 4 =(x- \hl{2} )(x+ \hl{2} )

Step 3 — Vertical asymptotes

Vertical asymptotes occur at x = 2 and x = negative 2.

x=2,x=−2x= \hl{2} ,\quad x=- \hl{2}

Step 4 — Compare degrees

Compare degrees: numerator 1 is less than denominator 2.

1<2\hl{1} < \hl{2}

Step 5 — Horizontal asymptote

Since numerator degree is smaller, the horizontal asymptote is y = 0.

y=0y= \hl{0}
rational-asymptotes For a rational function f(x) = p(x)/q(x):