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```					          Chapter 1

Limits and Their Properties:
Limits at Infinity

1
Limits at Infinity
   Remember: Limits at infinity do not actually
exist. Rather, it’s just an expression to say what
happens to y as x goes very, very far to the right
(or left for limits at negative infinity).
   The formal definition of a limit at infinity is
based on the formal definition of limit. This
definition is written in a way that makes it easy
for mathematician to do proofs, but in a way that
most students just get confused…
   Simply put, when we determine limits at
infinity, we are generally trying to find a
functions “end behavior”
Limits at Infinity
   Definition: The line y = L is a horizontal
asymptote of the graph of f if:
lim f ( x)  L or lim f ( x)  L
x                x 

   Graphical Example:              8
7
y

6
5
4
3
2
1            x

-1
-2
-3
-4
-5
-6
Limits at Infinity
   To analytically evaluate lim f ( x), we just need
x 
to rewrite the function as a Rational Expression
and follow a few simple rules. We will use the
following as our generic rational expression:
an x n  an1 x n1  ... a1 x  a0
f ( x)                  m 1
bm x  bm1 x  ... b1 x  b0
m

   The key parts of the function are n, m, an, and
bm. Everything else is insignificant.
Limits at Infinity
   Let’s try to better understand the following
an x n  an1 x n1  ... a1 x  a0
notation: f ( x)                  m 1
bm x  bm1 x  ... b1 x  b0
m

5 x  3x  2 x  11x  12
6      5      3
For example : f ( x) 
7 x  4 x  6 x  10 x  1
5     4     2

   What is n? n = 6               What is a5? a5 = -3
   What is m? m = 5               What is a4? a4 = 0
   What is a6? a6 = 5             What is a0? a0 = 12
   What is b5? b5 = 7             What is b0? b0 = -1
Limits at Infinity
an x n  an1 x n1  ... a1 x  a0
f ( x)                  m 1
bm x  bm1 x  ... b1 x  b0
m

   If you examine the degree of the numerator and
denominator, there’s one of three possibilities:
1.   n > m: The degree of the numerator is bigger than the
denominator (top heavy)
a) L =   There is no horizontal asymptote

2.   n < m: The degree of the numerator is smaller than
the denominator (bottom heavy)
a) L = 0     The horizontal asymptote is y = 0
3.   n = m: The degree of the numerator is equal to the
denominator
an                                       an
a) L =        The horizontal asymptote is y =
bm                                        bm
Limits at Infinity
   Are there any special limits to infinity for sine
and cosine?
   Of course there is!!!
   Luckily there’s not a lot that you need to
remember:
sin x                cos x
lim         0 or    lim        0
x    x             x    x
The horizontal asymptote is y = 0
   Just don’t get it confused with our limit as x
approaches zero!
sin x
lim        1
x 0   x
Limits at Infinity
   Example 1:
4 x  50
2
   Evaluate lim 3
x   x  85

   What is n? n = 2
   What is m? m = 3
   Which is bigger? m
   So, what’s the limit? L = 0
Limits at Infinity
   Example 2:
4 x  5 x  3x  1
3     2
   Evaluate lim
x    5 x 3  7 x  25
   What is n? n = 3
   What is m? m = 3
   Which is bigger? n = m
4
   So, what’s the limit? L =
5
Limits at Infinity
   Example 3:
3x  23
3
   Evaluate lim
x  4 x  1

   What is n? n = 3
   What is m? m = 1
   Which is bigger? n
   So, what’s the limit? L =   
Limits at Infinity
   Example 4:
x  23
3
   Evaluate lim
x    5 x 2  11 x  1

   What is n? n = 3
   What is m? m = 2
   Which is bigger? n
   So, what’s the limit? L =  
Limits at Infinity
   Example 5:
4 x  5x  3
2
   Evaluate lim
x  2 x 2  5 x  1

   What is n? n = 2
   What is m? m = 2
   Which is bigger? n = m
   So, what’s the limit? L =        2
Limits at Infinity
   Example 6:
4 x  5x  3
2
   Evaluate lim
x   2 x 2  5 x 3  1

   What is n? n = 2
   What is m? m = 3
   Which is bigger? m
   So, what’s the limit? L =          0
Limits at Infinity
   Example 7:
x  3x
2
   Evaluate lim
x    2x 1
   What is n? n = 1
   What is m? m = 1
   Which is bigger? n = m
1
   So, what’s the limit? L =
2
Limits at Infinity
   Example 8:
3x  3x
2
   Evaluate lim
x    2x 1
   What is n? n = 1
   What is m? m = 1
   Which is bigger? n = m
3
   So, what’s the limit? L = 
2
Limits at Infinity
   Example 9:
3x  11 x
3
   Evaluate lim
x   4 x 2  5 x

   What’s the limit? L =  

   Example 10:
6 x  11 x
3         2
   Evaluate lim
x   3 x 2  2 x 3

   What’s the limit? L =  3
Limits at Infinity
   Example 11:
sin 2 x       2 sin 2 x           sin 2 x
   Evaluate lim           lim            2  lim
x     x     x  2    x          x  2 x

   Now, let u = 2x
sin u
 2  lim        20  0
u  u

   Example 12:
1
   Evaluate lim cos 
x 
 x
   What’s the limit? L = 1
Homework
   Limits at Infinity
   Worksheet: Limits Analytically with Infinite Limits -
“What did the college freshman who failed
his first calculus test have in common with
the college freshman who was fined for
driving 60 mph in a 30 mph zone?”

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