infinite limits and limits to infinity: horizontal and vertical asymptotes
Post on 28-Dec-2015
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Recall…
• The notation tells us how the
limit fails to exist by denoting the unbounded behavior of f(x) as x approaches c.
• Infinity is not a number!
lim ( )x cf x
Properties of Infinite Limits
• Let c and L be real numbers and let f and g be functions such that
and
1. Sum or difference:
Consider:
lim ( )x cf x
lim ( )
x cg x L
lim[ ( ) ( )]x c
f x g x
2
1( )f x
x ( ) 2g x
0lim[ ( ) ( )]x
f x g x
0
lim ( )xf x
0lim ( )xg x
2
Properties of Infinite Limits
• Let c and L be real numbers and let f and g be functions such that
and
1. Product: if L > 0
if L < 0
Consider:
lim ( )x cf x
lim ( )
x cg x L
lim[ ( ) ( )]x c
f x g x
2
1( )f x
x ( ) 2g x
0lim[ ( ) ( )]x
f x g x
0
lim ( )xf x
0lim ( ) 2xg x
lim[ ( ) ( )]x c
f x g x
Properties of Infinite Limits
• Let c and L be real numbers and let f and g be functions such that
and
1. Quotient:
Consider:
lim ( )x cf x
lim ( )
x cg x L
( )lim
( )x c
g x
f x
2
1( )f x
x ( ) 2g x
0
( )lim
( )x
g x
f x
0lim ( )xf x
0lim ( ) 2xg x
0
0
Definition - Vertical Asymptotes
• If f(x) approaches infinity (or negative infinity) as x approaches c from the left or the right, then the line x = c is a vertical asymptote of the graph of f.
verticalasymptote
Determining Infinite Limits
3( )f x
x
2
4( )g x
x
3
2( )h x
x
0lim ( )x
f x
0lim ( )x
f x
0lim ( )x
g x
0lim ( )x
g x
0lim ( )x
h x
0lim ( )x
h x
The pattern…
Is c even or odd?
Sign of
p(x) when
x = c
odd positive
odd negative
even positive
even negative
0lim ( )x
f x 0
lim ( )x
f x
( )( ) , where ( ) is a polynomial
c
p xf x p x
x
and c is a positive integer
Using the pattern…
0
3limx
x
x
3
20
2 3limx
x x
x
50
2 1limx
x
x
0
3limx
x
x
3
20
2 3limx
x x
x
50
2 1limx
x
x
Using the pattern…
6
100
3 6limx
x
x
2
20
3limx
x x
x
6
100
3 6limx
x
x
2
20
3limx
x x
x
0
3limx
x
x
0
3limx
x
x
Limits at Infinity
• denotes that as x
increases without bound, the function value approaches L
• L can have a numerical value, or the limit can be infinite if f(x) increases (decreases) without bound as x increases without bound
lim ( )x
f x L
Horizontal Asymptotes
• The line y = L is a horizontal asymptote of f if
or
• Notice that a function can have at most two HORIZONTAL asymptotes (Why?)
lim ( )x
f x L
lim ( )x
f x L
4
2
-2
-4
-5 5
Note: It IS possible for a graph to cross its horizontal asymptote!!!!!!
lim ( ) ______x
f x
lim ( ) ______x
f x
2 2
Horizontal Asymptote(s):__________
lim ( ) ______x
f x
lim ( ) ______x
f x
0 0
Horizontal Asymptote(s):__________
8
6
4
2
-2
-10 -5 5 10
Theorem – Limits at Infinity
1. If r is a positive rational number and c is any real number, then
The second limit is valid only if xr is defined when x < 0
limrx
c
x lim
rx
c
x
lim x
xe
lim x
xe
0
00
0
Using the Theorem
5limx x
13
5limx x
5lim 2x x
lim x
xe
lim 2 x
xe
0 0
2
0 0
5lim lim 2x xx
lim 2 lim x
x xe
Guidelines for Finding Limits at ±∞ of Rational Functions
1. If the degree of the numerator is ___________ the degree of the denominator, then the limit of the rational function is ___.
2. If the degree of the numerator is _______ the degree of the denominator, then the limit of the rational function is the __________________ _______________________.
3. If the degree of the numerator is ___________ the degree of the denominator, then the limit of the rational function _______________.
greater than
0
less than
equal to
the ratio of the leading coefficients
is infinite
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