Mathematics (from Greek μάθημα máthēma, “knowledge, study, learning”) is the study of topics such as quantity (numbers), structure, space, and change. There is a range of views among mathematicians and philosophers as to the exact scope and definition of mathematics

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!