Section 2.2: Infinite Limits and Limits at Infinity

Infinite Limits

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^-

Vertical Asymptotes

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.]

Vertical Asymptotes of Rational Functions

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.

Limits at Infinity

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

Limits of Power Functions at Infinity

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

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

Limits of Polynomial Functions at Infinity

If

p(x) = a_nxⁿ + a_{n - 1}x^{n - 1} + ... + a₁x + a₀, a_n ≠ 0, n ≥ 1

then

limx → ∞ p(x) = limx → ∞ a_nxⁿ = ± ∞andlimx → -∞ p(x) = limx → -∞ a_nxⁿ = ± ∞

Each limit will be either -∞ or ∞, depending on a_n and n.

Limits of Rational Functions at Infinity

If

f(x) = [a_mx^m + a_{m - 1}x^{m - 1} + ... + a₁x + a₀]/[b_nxⁿ + b_{n - 1}x^{n - 1} + ... + b₁x + b₀], a_m ≠ 0, b_n ≠ 0

then

limx → ∞f(x) = limx → ∞a_mx^m/b_nxⁿandlimx → -∞f(x) = limx → -∞a_mx^m/b_nxⁿ

Horizontal Asymptotes of Rational Functions

There are three possible cases for these limits:

1. 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).
2. 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).
3. If m > n, then each limit will be ∞ or -∞, depending on m, n, a_m, and b_n, and f(x) does not have a horizontal asymptote.