Finding vertical, horizontal and slant asymptotes with limits, telling holes from vertical asymptotes, and end behavior of functions with radicals.
What an asymptote is
An asymptote is a line that a graph approaches more and more closely as \(x\) or \(y\) heads off to infinity. Asymptotes describe a graph's long-range shape, and limits are how you find them.
Vertical asymptotes
For a rational function, look where the denominator is zero. But check the numerator too.
Horizontal asymptotes
A graph may cross a horizontal asymptote; what matters is the behavior far out.
Slant (oblique) asymptotes
When the degree of the numerator is exactly one more than the degree of the denominator, the graph approaches a slanted line. Find it with polynomial division: the quotient is the asymptote.
Practice
Quick check
What is the horizontal asymptote of \(f(x) = \frac{4x + 7}{2x - 1}\)?