relative rates of growth section 8.2. the exponential function grows so rapidly and the natural...
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Relative Rates of GrowthSection 8.2
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The exponential function grows so rapidlyand the natural logarithm function growsso slowly that they set standards by which wecan judge the growth of other functions...
xeln x
Comparing Rates of Growth
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As an illustration of how rapidly grows, imagine graphing the function on a boardwith the axes labeled in centimeters…
xe
At x = 1 cm, the graph is cm high.1 3e
At x = 6 cm, the graph is m high.6 4e
At x = 10 cm, the graph is m high.10 220e
At x = 24 cm, the graph is more than half wayto the moon.
At x = 43 cm, the graph is light-yearshigh (well past Proxima Centauri, the neareststar to the Sun).
43 5.0e
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Let f (x) and g(x) be positive for x sufficiently large.
Faster, Slower, Same-rate Growth as x
1. f grows faster than g (and g grows slower than f )as ifx
limx
f x
g x or, equivalently, if
lim 0x
g x
f x
2. f and g grow at the same rate as ifx
lim 0x
f xL
g x (L finite and not zero)
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According to these definitions, does notgrow faster than as . The twofunctions grow at the same rate because
Faster, Slower, Same-rate Growth as x 2y x
y x x
2lim lim 2 2x x
x
x
which is a finite nonzero limit. The reason for thisapparent disregard of common sense is that we want“f grows faster than g” to mean that for largex-values, g is negligible in comparison to f.
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If f grows at the same rate as g as and ggrows at the same rate as h as , then f growsat the same rate as h as .
Transitivity of Growing Ratesx
x x
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Determine whether the given function grows faster than,at the same rate as, or slower than the exponentialfunction as x approaches infinity.
Guided Practice
5
2
x
Our new rule: 5 2
limx
xx e
5lim
2
x
x e
0 (because the baseis less than one!)
Grows slower than as x xe
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Determine whether the given function grows faster than,at the same rate as, or slower than the exponentialfunction as x approaches infinity.
Guided Practice
lnx x xln
limxx
x x x
e
Grows slower than as x xe
1 ln 1lim
xx
x x x
e
lnlim
xx
x
e
1lim
xx
x
e
00
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Determine whether the given function grows faster than,at the same rate as, or slower than the squaring functionas x approaches infinity.
Guided Practice
3 3x 3
2
3limx
x
x
Grows faster than as x 2x
2
3limx
xx
0
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Determine whether the given function grows faster than,at the same rate as, or slower than the squaring functionas x approaches infinity.
Guided Practice
4 5x x4
2
5limx
x x
x
Grows at the same rate as as x 2x
4
4
5limx
x x
x
3
5lim 1x x
1 0 1
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Determine whether the given function grows faster than,at the same rate as, or slower than the squaring functionas x approaches infinity.
Guided Practice
2x
2
2lim
x
x x
Grows faster than as x 2x
ln 2 2lim
2
x
x x
2ln 2 2
lim2
x
x
2
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Determine whether the given function grows faster than,at the same rate as, or slower than the natural logarithmfunction as x approaches infinity.
Guided Practice
log x
loglim
lnx
x
x
Grows at the same rate as as x ln x
loglim
2lnx
x
x
ln ln10lim
2lnx
x
x
1
2ln10
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Show that the three functions grow at the same rate asx approaches infinity.
Guided Practice
31f x x
4 2
2
2 1
1
x xf x
x
5
3 2
2 1
1
xf x
x
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Compare the first and second functions:
2
1
limx
f x
f x
4 2
3
2 11lim
x
x xxx
4 2
4 3
2 1limx
x x
x x
1
11
Rational Function Theorem!
Compare the first and third functions:
3
1
limx
f x
f x
5
2
3
2 11lim
x
xx
x
5
5 3
2 1limx
x
x x
2
21
By transitivity, the second and third functions grow at thesame rate, so all three functions grow at the same rate!