Rational Functions and Asymptotes

Asymptotes of rational functions

A rational function $f(x) = p(x)/q(x)$ has two types of asymptotes:

Vertical asymptote (VA)

Set the denominator equal to zero: $q(x) = 0$. The function blows up to $\pm\infty$ at these values. Example: $f(x) = \frac{x+1}{x-2}$ has VA at $x = 2$.

Horizontal asymptote (HA)

Compare the degree of numerator vs denominator:

• Same degree → HA = ratio of leading coefficients.
• Numerator lower → HA is $y = 0$.
• Numerator higher → no HA (oblique/slant asymptote instead).

Example: $\frac{x+1}{x-2}$: both degree 1 → HA at $y = 1/1 = 1$.

Intercepts

x-intercept: set numerator $= 0$. y-intercept: compute $f(0)$.

The graph behavior

The curve approaches but never crosses the asymptotes (usually). Near the VA, the function shoots to $\pm\infty$. Far from the origin, it flattens toward the HA.

Try it in the visualization

Adjust the numerator and denominator. VA and HA update automatically. The curve shows the approaching behavior. $q(x) = 0$; horizontal asymptote determined by comparing degrees of $p$ and $q$. For $(x+1)/(x-2)$: VA at $x=2$, HA at $y=1$ (same degree → ratio of leading coefficients).