1. 1.5 Infinite Limits

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

3. Infinite Limits
Graphically

4. 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.

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

6. Example

7. Example

8. Examples

9. 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.

10. 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).

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

12. 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:

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

14. Example

15. Example

16. Example

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