# 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.

x = 2, f(x) = 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/xp = 0
2. 2.limx → ∞ k/xp = 0
3. 3.limx → -∞ kxp = ± ∞
4. 4.limx → ∞ kxp = ± ∞

provided that xp 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) = kxp as k and p change.

### Limits of Polynomial Functions at Infinity

If

p(x) = anxn + an - 1xn - 1 + ... + a1x + a0, an ≠ 0, n ≥ 1

then

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

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

k = 1

p = 2

## Limits of Rational Functions at Infinity

If

f(x) = [amxm + am - 1xm - 1 + ... + a1x + a0]/[bnxn + bn - 1xn - 1 + ... + b1x + b0], am ≠ 0, bn ≠ 0

then

limx → ∞f(x) = limx → ∞amxm/bnxnandlimx → -∞f(x) = limx → -∞amxm/bnxn

### 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) = am/bn, and the line y = am/bn is a horizontal asymptote of f(x).
3. If m > n, then each limit will be ∞ or -∞, depending on m, n, am, and bn, and f(x) does not have a horizontal asymptote.