Calculus Chapter One

1. Calculus Chapter One Sec 1.5 Infinite Limits

2. 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.

3. 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=1 • The limits equal the height of the horizontal asymptote and are written as lim f(x) = 1 and lim f(x) = 1 x→∞ x→-∞

4. 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.

5. 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) = 0 x→∞ 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.

6. 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.

7. Sec 1.5 • Solving limits at infinity with a calculator.

8. Sec 1.5 • Using algebra for limits at infinity.