Infinite Limits and Limits to Infinity: Horizontal and
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Transcript Infinite Limits and Limits to Infinity: Horizontal and
Infinite Limits and
Limits to Infinity: Horizontal
and Vertical Asymptotes
Recall…
• The notation
lim f ( x)
x c
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!
Properties of Infinite Limits
•
Let c and L be real numbers and let f and g be
functions such that
lim f ( x) and lim g ( x ) L
x c
xc
f ( x) g ( x)]
1. Sum or difference: lim[
x c
1
Consider: f ( x) x 2
lim[ f ( x) g ( x)]
x 0
g ( x) 2
lim f ( x )
lim g ( x )
2
x0
x0
Properties of Infinite Limits
•
Let c and L be real numbers and let f and g be
functions such that
lim f ( x) and lim g ( x ) L
x c
xc
f ( x) g ( x)]
1. Product: if L > 0 lim[
x c
if L < 0 lim[ f ( x) g ( x)]
x c
1
Consider: f ( x) x 2
lim[ f ( x) g ( x)]
x 0
g ( x) 2
lim f ( x)
x 0
lim g ( x ) 2
x 0
Properties of Infinite Limits
•
Let c and L be real numbers and let f and g be
functions such that
lim f ( x) and lim g ( x ) L
x c
xc
1. Quotient: lim g ( x)
x c f ( x)
0
1
Consider: f ( x) x 2
g ( x)
lim
x 0
f ( x)
0
g ( x) 2
lim f ( x)
x 0
lim g ( x ) 2
x 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.
Determining Infinite Limits
3
f ( x)
x
4
g ( x) 2
x
2
h( x ) 3
x
lim f ( x)
x 0
lim f ( x)
x 0
lim g ( x)
x 0
lim g ( x )
x 0
lim h( x )
x 0
lim h( x )
x0
The pattern…
Is c even or
odd?
p( x)
f ( x) c , where p( x) is a polynomial
x and c is a positive integer
odd
Sign of
p(x) when
x=c
positive
odd
negative
even
even
lim f ( x)
x 0
lim f ( x)
x 0
positive
negative
Using the pattern…
3 x
lim
x 0
x
x3 2 x 3
lim
2
x 0
x
2x 1
lim
5
x 0
x
3 x
lim
x 0
x
x3 2 x 3
lim
2
x 0
x
2x 1
lim
5
x 0
x
Using the pattern…
3x 6 6
lim
10
x 0
x
x 2 3x
lim
x 0
x2
x3
lim
x 0
x
3x 6 6
lim
10
x 0
x
x 2 3x
lim
2
x 0
x
x3
lim
x 0
x
Limits at Infinity
• lim f ( x) L denotes that as x
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
Horizontal Asymptotes
• The line y = L is a horizontal asymptote of f if
lim f ( x) L
x
or
lim f ( x) L
x
• Notice that a function can have at most two
HORIZONTAL asymptotes (Why?)
4
2
-5
lim f ( x) ______
0
5
-2
x
lim f ( x) ______
0
x
-4
Horizontal Asymptote(s):__________
4
2
-5
lim f ( x) ______
2
5
-2
x
lim f ( x) ______
2
x
-4
Note: It IS possible for a graph to cross
its horizontal asymptote!!!!!!
Horizontal Asymptote(s):__________
4
2
-5
lim f ( x) ______
0
x
5
-2
lim f ( x) ______
1
x
Horizontal Asymptote(s):__________
-4
4
2
-10
-5
5
10
-2
lim f ( x) ______
0
x
lim f ( x) ______
0
x
Horizontal Asymptote(s):__________
Theorem – Limits at Infinity
1. If r is a positive rational number and c is
any real number, then
c
lim r
x x
0
c
lim r
x x
0
The second limit is valid only if xr is
defined when x < 0
lim e 0
x
x
lim e
x
x
0
Using the Theorem
5
lim 13
x x
5
lim 0
x x
5
5
lim 2 lim lim 2
x x
x x
x
lim e
x
x
0
0
2
lim 2e lim 2 lim e 0
x x
x
x
x
Guidelines for Finding Limits at
±∞ of Rational Functions
less than
1. If the degree of the numerator is ___________
the degree of the denominator, then the limit of
the rational function is ___.
0
2. If the degree of the numerator is equal
_______
to the
degree of the denominator, then the limit of the
the ratio of the
rational function is the __________________
leading coefficients
_______________________.
than
3. If the degree of the numerator is greater
___________
the degree of the denominator, then the limit of
is infinite
the rational function _______________.
Using the Guidelines…
x3 2 x 3
lim 4
0
2
x x 3 x x
6 x8 12 x 17
lim
x 18 x8 13 x 2 24
1
3
2 x3 9 x 3
lim 3
x x 7 x 2 1
2
x5 5 x 6
lim 3
x x 2 x 2 8
∞