In the example on the right, we write:

limx → 1

^{+}f(x) = ∞orf(x) → ∞asx → 1^{+}

and

limx → 1

^{-}f(x) = -∞orf(x) → -∞asx → 1^{-}

The vertical line *x = a* is a **vertical asymptote** for the graph of *y = f(x)* if

f(x) → ∞orf(x) → -∞asx → a

^{+}orx → a^{-}.

[That is, *f(x)* either increases or decreases without bound as *x* approaches *a* from the right or from the left.]

If f(x) = n(x)/d(x) is a rational function, d(c) = 0 and n(c) ≠ 0, then the line x = c is a vertical asymptote of the graph of *f*.

Drag the slider to see the value of the function f(x) = 1/(x - 1) as *x* is close to 1.

A line y = b is a **horizontal asymptote** of the graph of *y = f(x)* if *f(x)* approaches *b* as either *x* increases without bound or *x* decreases without bound. We write this as:

limx → -∞ f(x) = borlimx → ∞ f(x) = b

If *p* is a positive real number and *k* is any real number except 0, then:

**1.**limx → -∞ k/x^{p}= 0**2.**limx → ∞ k/x^{p}= 0**3.**limx → -∞ kx^{p}= ± ∞**4.**limx → ∞ kx^{p}= ± ∞

provided that x^{p} is a real number for negative values of *x*. The limits in 3 and 4 with be either -∞ or ∞, depending on *k* and *p*.

Drag the slider to see the graph of the function f(x) = kx^{p} as *k* and *p* change.

If

p(x) = a

_{n}x^{n}+ a_{n - 1}x^{n - 1}+ ... + a_{1}x + a_{0}, a_{n}≠ 0, n ≥ 1

then

limx → ∞ p(x) = limx → ∞ a

_{n}x^{n}= ± ∞andlimx → -∞ p(x) = limx → -∞ a_{n}x^{n}= ± ∞

Each limit will be either -∞ or ∞, depending on a_{n} and *n*.

If

f(x) = [a

_{m}x^{m}+ a_{m - 1}x^{m - 1}+ ... + a_{1}x + a_{0}]/[b_{n}x^{n}+ b_{n - 1}x^{n - 1}+ ... + b_{1}x + b_{0}], a_{m}≠ 0, b_{n}≠ 0

then

limx → ∞f(x) = limx → ∞a

_{m}x^{m}/b_{n}x^{n}andlimx → -∞f(x) = limx → -∞a_{m}x^{m}/b_{n}x^{n}

There are three possible cases for these limits:

- If
*m < n*, then limx → ∞f(x) = limx → -∞f(x) = 0, and the line y = 0 (the*x*axis) is a horizontal asymptote of f(x). - If
*m = n*, then limx → ∞f(x) = limx → -∞f(x) = a_{m}/b_{n}, and the line y = a_{m}/b_{n}is a horizontal asymptote of f(x). - If
*m > n*, then each limit will be ∞ or -∞, depending on*m*,*n*,*a*, and_{m}*b*, and_{n}*f(x)*does not have a horizontal asymptote.