Asymptotes

An asymptote for the curve $y=f(x)$ is a straight line which is, informally, "a tangent at infinity" to the curve — the distance between the curve and the line tends to $0$ as the point on the curve runs off to infinity. There are three types.

1. Horizontal asymptote (parallel to the $x$-axis). $y=L$ is a horizontal asymptote for $y=f(x)$ if either

2. Vertical asymptote (parallel to the $y$-axis). $x=a$ is a vertical asymptote for $y=f(x)$ if

3. Slant (oblique) asymptote. Occurs when, for a rational function, the numerator's degree is exactly one higher than the denominator's. Found by dividing the numerator by the denominator (long or synthetic division): the quotient (a linear polynomial) is the slant asymptote; the remainder over the denominator vanishes as $x\to\pm\infty$.

Worked patterns (Examples 7.66–7.68):

• $f(x)=\dfrac1x$: as $x\to0^-$, $f\to-\infty$; as $x\to0^+$, $f\to+\infty$ — so $x=0$ is a vertical asymptote; by symmetry (the curve is unchanged under $x\leftrightarrow y$), $y=0$ is a horizontal asymptote too, giving the familiar rectangular-hyperbola shape.
• $\dfrac{x^2-6x+7}{x+5}$-type expression: numerator degree (2) exceeds denominator degree (1) by exactly one $\Rightarrow$ a slant asymptote exists; long division gives the linear quotient directly as the asymptote's equation (the curve approaches this line but never actually touches it). …