The document covers concepts related to infinite limits and limits at infinity in AP Calculus, discussing vertical asymptotes and identifying infinite limits through graphical behavior. It includes examples and techniques for handling indeterminate forms and determining vertical asymptotes in rational functions. Additionally, it outlines properties of infinite limits and the use of horizontal asymptotes, along with word problems applying these concepts.

1. 1.5 Infinite Limits
and 3.5 Limits at Infinity
AP Calculus I
Ms. Hernandez
(print in grayscale or black/white)

2. AP Prep Questions / Warm Up
No Calculator!
(a) 1 (b) 0 (c) e (d) –e (e) Nonexistent
(a) –1/4 (b) –1/2 (c) 0 (d) 1 (e) DNE
1
ln
lim
x
x
x→
22
( 2)
lim
4x
x
x→−
+
−

3. AP Prep Questions / Warm Up
No Calculator!
(a) 1 (b) 0 (c) e (d) –e (e) Nonexistent
(a) –1/4 (b) –1/2 (c) 0 (d) 1 (e) DNE
1
ln ln1 0
lim 0
1 1x
x
x→
= = =
22 2 2
( 2) ( 2) 1 1
lim lim lim
4 ( 2)( 2) ( 2) 4x x x
x x
x x x x→− →− →−
+ +
= = = −
− + − −

4. 1.5 Infinite Limits
 Vertical asymptotes at x=c will give you
infinite limits
 Take the limit at x=c and the behavior of
the graph at x=c is a vertical asymptote
then the limit is infinity
 Really the limit does not exist, and that
it fails to exist is b/c of the unbounded
behavior (and we call it infinity)

5. Determining Infinite Limits from a
Graph
 Example 1 pg 81
 Can you get different infinite limits from
the left or right of a graph?
 How do you find the vertical asymptote?

6. Finding Vertical Asymptotes
 Ex 2 pg 82
 Denominator = 0 at x = c AND the
numerator is NOT zero
 Thus, we have vertical asymptote at x = c
 What happens when both num and den
are BOTH Zero?!?!

7. A Rational Function with Common
Factors
 When both num and den are both zero then
we get an indeterminate form and we have to
do something else …
 Ex 3 pg 83
 Direct sub yields 0/0 or indeterminate form
 We simplify to find vertical asymptotes but how do
we solve the limit? When we simplify we still have
indeterminate form.
2
22
2 8
lim
4x
x x
x→−
+ −
−
2
4
lim , 2
2x
x
x
x→−
+
≠ −
+

8. A Rational Function with Common
Factors
 Ex 3 pg 83: Direct sub yields 0/0 or
indeterminate form. When we simplify
we still have indeterminate form and we
learn that there is a vertical asymptote
at x = -2.
 Take lim as x-2 from left and right
2
2
2
2 8
lim
4x
x x
x+
→−
+ −
−
2
2
2
2 8
lim
4x
x x
x−
→−
+ −
−

9. A Rational Function with Common
Factors
 Ex 3 pg 83: Direct sub yields 0/0 or indeterminate
form. When we simplify we still have indeterminate
form and we learn that there is a vertical asymptote
at x = -2.
 Take lim as x-2 from left and right
 Take values close to –2 from the right and values
close to –2 from the left … Table and you will see
values go to positive or negative infinity
2
2
2
2 8
lim
4x
x x
x+
→−
+ −
= ∞
−
2
2
2
2 8
lim
4x
x x
x−
→−
+ −
= −∞
−

10. Determining Infinite Limits
 Ex 4 pg 83
 Denominator = 0 when x = 1 AND the
numerator is NOT zero
 Thus, we have vertical asymptote at x=1
 But is the limit +infinity or –infinity?
 Let x = small values close to c
 Use your calculator to make sure – but
they are not always your best friend!

11. Properties of Infinite Limits
 Page 84
 Sum/difference
 Product L>0, L<0
 Quotient (#/infinity = 0)
 Same properties for
 Ex 5 pg 84
lim ( )
x c
f x
→
= ∞
lim ( )
x c
g x L
→
=
lim ( )
x c
f x
→
= −∞

12. Asymptotes & Limits at Infinity
For the function , find
(a)
(b)
(c)
(d)
(e) All horizontal asymptotes
(f) All vertical asymptotes
2 1
( )
x
f x
x
−
=
lim ( )
x
f x
→∞
lim ( )
x
f x
→−∞
0
lim ( )
x
f x+
→
0
lim ( )
x
f x−
→

13. Asymptotes & Limits at Infinity
For x>0, |x|=x (or my x-values are positive)
1/big = little and 1/little = big
sign of denominator leads answer
For x<0 |x|=-x (or my x-values are negative)
2 and –2 are HORIZONTAL Asymptotes
2 1
( )
x
f x
x
−
=
2 1 2 1 1
lim ( ) lim lim lim 2 2
x x x x
x x
f x
x x x→∞ →∞ →∞ →∞
− −  
= = = − = ÷
 
2 1 2 1 1
lim ( ) lim lim lim 2 2
x x x x
x x
f x
x x x→−∞ →−∞ →−∞ →∞
− −  
= = = − + = − ÷
−  

14. Asymptotes & Limits at Infinity
2 1
( )
x
f x
x
−
=
0 0 0 0
2 1 2 1 1
lim ( ) lim lim lim 2
x x x x
x x
f x
x x x+ + + +
→ → → →
− −  
= = = − ÷
 
2 1 2 1 1
lim ( ) lim lim lim 2 2
x x x x
x x
f x
x x x→−∞ →−∞ →−∞ →∞
− −  
= = = − + = − ÷
−  
1 1
2 2 2 lim DNE
x little
   
− = − = −∞ = −∞ ∴ ÷  ÷
+   
1 1
2 2 2 lim DNE
x little
   
− + = − + = − +−∞ = −∞ ∴ ÷  ÷
−   

15. 3.5 Limit at Infinity
 Horizontal asymptotes!
 Lim as xinfinity of f(x) = horizontal
asymptote
 #/infinity = 0
 Infinity/infinity
 Divide the numerator & denominator by a
denominator degree of x

16. Some examples
 Ex 2-3 on pages #194-195
 What’s the graph look like on Ex 3.c
 Called oblique asymptotes (not in cal 1)
 KNOW Guidelines on page 195

17. 2 horizontal asymptotes
 Ex 4 pg 196
 Is the method for solving lim of f(x) with
2 horizontal asymptotes any different
than if the f(x) only had 1 horizontal
asymptotes?

18. Trig f(x)
 Ex 5 pg 197
 What is the difference in the behaviors
of the two trig f(x) in this example?
 Oscillating toward no value vs
oscillating toward a value

19. Word Problems !!!!!
 Taking information from a word problem
and apply properties of limits at infinity
to solve
 Ex 6 pg 197

20. A word on infinite limits at infinity
 Take a lim of f(x)  infinity and
sometimes the answer is infinity
 Ex 7 on page 198
 Uses property of f(x)
 Ex 8 on page 198
 Uses LONG division of polynomials-Yuck!