1.5 Infinite Limits

Objectives Determine infinite limits from the left and from the right. Find and sketch the vertical asymptotes of the graph of a function.

Infinite Limits Graphically

Infinite Limits Analytically: Plug in number. If you get # / 0, you know it’s either ∞ or -∞. Check sign by plugging in a number close on the appropriate side.

Infinite Limits If the function increases without bound, the limit is +∞. If the function decreases without bound, the limit is -∞.

Example

Example

Examples

As x approaches 1, the graphs become arbitrarily close to the vertical line x=1. This line is called a vertical asymptote. If f(x) approaches ∞ or -∞, as x approaches c from the right or from the left, then the line x=c is a vertical asymptote of the graph of f.

Theorem 1.14 Let f and g be continuous on an open interval containing c. If f(c)≠0, g(c)=0 and there exists an open interval containing c such that g(x)≠0 for all x≠c in the interval, then the graph of the function given by h(x)=f(x) / g(x) has a vertical asymptote at x=c. (Vertical asymptotes occur at numbers that make the denominator 0, but NOT the numerator).

Vertical Asymptotes Find all the vertical asymptotes: x=2 x=3 x= -2

Properties of Infinite Limits Theorem 1.15 Properties of Infinite Limits Let c and L be real numbers and let f and g be functions such that sum/difference: product: quotient:

What do you think? ∞ + ∞ = ∞ -∞ - ∞ = -∞ # / ∞ = 0 ∞ - ∞ = ???

Example

Example

Example

Homework 1.5 (page 85) #1, 3, 9-51 odd (Don’t graph) Handout (2.5) #39, 47, 51