1. Section 1.5: Infinite Limits

2. Vertical Asymptote
If f(x) approaches infinity (or negative infinity) as x approaches c from the right or the left, then the line x = c is a vertical asymptote of the graph of f.

3. Infinite Limits The limit statement such as
means that the function f increases without bound as x approaches c from either side, while
means that the function g decreases without bound as x approaches c from either side.

4. Example 1
Sketch a graph of a function with the following characteristics: The graph has discontinuities at x = -2, 0, and 3. Only x = 0 is removable.

5. Example 2
Use the graph and complete the table to find the limit (if it exists). x
1.9
1.99
1.999 2
2.001
2.01
2.1
f(x)
-100
-10000
DNE
-10000
-100
If the function behaves the same around an asymptote, then the infinite limit exists.
The function decreases without bound as x approaches 2 from either side.

6. Example 3
Use the graph and complete the table to find the limit (if it exists). x
1.9
1.99
1.999 2
2.001
2.01
2.1
f(x)
-10
-100
-1000
DNE
1000
100 10
If the function behaves the different around an asymptote, then the infinite limit does not exist.
The function increases without bound as x approaches 2 from the right and decreases without bound as x approaches 2 from the left.

7. One-Sided Infinite Limits do Exist
Example 4
Use the graph and complete the table to find the limits (if they exist). x
1.9
1.99
1.999 2
2.001
2.01
2.1
f(x)
-10
-100
-1000
DNE
1000
100 10
One-Sided Infinite Limits do Exist
The function increases without bound as x approaches 2 from the right and decreases without bound as x approaches 2 from the left.

8. The Existence of a Vertical Asymptote
If is continuous c around and g(x) ≠ 0 around c, then x = c is a vertical asymptote of h(x) if f(c) ≠ 0 and g(c) = 0.
Big Idea: x = c is a vertical asymptote if c ONLY makes the denominator zero.
Ex: Determine all vertical asymptotes of
When is the denominator zero:
Do the x’s make the numerator 0?
No for both
x=1 and x=-1 are vertical asymptotes
Must have equations for asymptotes

9. Example 2 Determine all vertical asymptotes of .
When is the denominator zero:
Do the x’s make the numerator 0?
Yes…
No!
x=2 is a vertical asymptote
EXTRA: What about x = -1?
Therefore, x=1 is a removable discontinuity

10. Example 3 Analytically determine all vertical asymptotes of
We Know:
When is the denominator zero:
Do the x’s make the numerator 0?
No, since the numerator is a constant.
0 and π are angles that make sine 0
Find all of the values since trig functions are cyclic

11. Example 3 Cont. Analytically determine all vertical asymptotes of
Check with the graph

12. Properties of Infinite Limits
Let c and L be real numbers and f and g be functions such that:
Sum/Difference:
Product:
Quotient:
Example: Since , then