Calculus Chapter One

Sec 1.5

Infinite Limits

Sec 1.5

Up until now, we have been looking at limits where x approaches a regular, finite number.

But x can also approach ∞ or -∞. Limits at infinity exist when a function

has a horizontal asymptote.

Sec 1.5

Consider the following function f(x)=(x+2) (x-5)

(x-3) (x+1)f(x) has a horizontal asymptote at y=1The limits equal the height of the

horizontal asymptote and are written as lim f(x) = 1 and lim f(x) = 1

x→∞ x→-∞

Sec 1.5

Horizontal asymptotes and limits at infinity always go hand in hand. You can’t have one without the other.

Suppose you have a rational function like f(x) = (3x-7)/(2x+8)

determining the limit at infinity or negative infinity is the same as finding the location of the horizontal asymptote.

Sec 1.5

First, note the degree of the numerator and the degree of the denominator.

If the degree of the numerator is greater than the degree of the denominator, there is NO horizontal asymptote and the limit of the function as x approaches infinity (or negative infinity) does not exist.

If the degree of the denominator is greater than the degree of the numerator, the x-axis is the horizontal asymptote and

lim g(x) = lim g(x) = 0x→∞ x-∞

If the degrees of the numerator and denominator are equal, take the coefficient of the highest power of x in the numerator and divide it by the coefficient of the highest power of x in the denominator. That quotient gives you the answer to the limit problem and the height of the asymptote. See example.

Sec 1.5

Substitution will not work for problems in this section.

If you try plugging ∞ into x in any of the rational functions in this section, you get ∞/∞, but that does NOT equal 1. A result of ∞/∞ tells you nothing about the answer to a limit problem.

Sec 1.5

Solving limits at infinity with a calculator.

Sec 1.5

Using algebra for limits at infinity.