1 5 Infinite Limits Objectives Determine infinite limits

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

Theorem 1. 15 Properties of Infinite Limits Let c and L be real numbers and let f and g be functions such that 1. sum/difference: 2. product: 3. quotient:

Example

Example

Example

Homework 1. 5 (page 85) #1, 3, 9 -51 odd (Don’t graph) Handout (2. 5) #11 -19 odd #39, 47, 51