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Title: ‘Relative Rates of Growth’
Goal: Compare the growth rates of some common and not so common functions.
Technology Requirements: Evaluate limits
Exposition:
All the functions we will look at in this lab grow as x → ∞, that is for each function f (x) that
we look at lim f (x) = ∞. However, not all functions grow at the same rate. We compare
x→∞
the growth rates of two functions by looking at the limit of their quotient as
x → ∞. Consider the following three limits. (An L over an equal sign indicates that we
have applied l’Hopital’s Rule.)
5
5x+1
• lim 3x
2 +2 = lim 6x = 0
x→∞
x→∞
L
2
L
3
L
+1
5
• lim 5x
= lim 10x
= lim 10
= 10
2
6 = 3
x→∞ 3x +2
x→∞ 6x
x→∞ 6
2
+1
15x
15x
• lim 5x
2 +2 = lim 6x = lim 6 = ∞
3x
x→∞
x→∞
x→∞
The first limit is 0 so the quadratic function g(x) = 3x2 + 2 grows larger faster than the
linear function f (x) = 5x + 1, so we say that the quadratic dominates the linear. That
is g(x) gets bigger faster and in time as x gets larger g will have a greater value than f .
The second limit is 5/3, this tells us that these two quadratic functions grow at the same
rate, neither one dominates. Finally, the last limit is infinity and so we know that the cubic
function dominates the quadratic. In general:
f (x)
• if lim g(x) = 0 then g(x) dominates f (x).
x→∞
f (x)
• if lim g(x) = L where 0 < |L| < ∞ then they grow at the same rate.
x→∞
f (x)
• if lim g(x) = ∞ then f (x) dominates g(x).
x→∞
Pre-Lab Questions:
Answer the following based on the exposition:
1. How do we compare the growth rates of two functions if they are both unbounded
(going to ∞) as x increases?
2. What does it mean for a function f (x) to grow faster than, or dominate, another
function g(x)?
3. How can we tell if one function dominates another?
4. If we want to compare the growth rates of f (x) = x2 to g(x) = 2x how do we do that?
5. What is the result of the comparison in the previous question?
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Now use these ideas to complete the lab.
Part 1:
Simplify each quotient if possible and then evaluate the limit by hand, use l’Hopital’s rule if
necessary.
ln(x)
• lim x1/2
x→∞
3
• lim xx1/2
x→∞
3
• lim xex
x→∞
1. For each limit indicate which function grows faster.
2. If you were to compare ln(x) to x3 which do you think would grow faster and why?
√
3. How do you think ln(x) and x each compare to ex and why?
Using what you have just done and more limits if necessary order the functions
√
ln(x), ex , x5 , x3/5 , x3 , x and x
from slowest growing to fastest growing. Be sure to give supporting evidence for your ordering. (Again do this by hand without using the computer.)
Part 2:
Only using your answers from Part 1 and no limits, try to order the functions
ln(x2 ), x3/2 , x20 + 300x + 100, x3 + x2 − x + 1, 5x, and ex
from slowest growing to fastest growing by comparing them to the functions in the previous
part. Give reasons for your conclusions. (Again do this by hand without the using the
computer.)
Part 3:
Finally using the computer to evaluate any necessary limits order the functions
π x , 3x , xπ , ex , xx , and x!
from slowest growing to fastest growing. Remember to give reasons for your conclusions.
(Note that x! is x factorial, e.g. 5! = 5 · 4 · 3 · 2 · 1 = 120, I’m not just really excited about
the function x.)
Extra Credit: Extra credit will be given for complete labs that use a minimum of limits.