Expert: Paul Klarreich - 10/18/2007

Question
Using the sandwich theorem, show that lim x->0 x^4(1-cosx)=0

Answer
Questioner:   Maria
Category:  Calculus
Private:  No
 
Subject:  question
Question:  Using the sandwich theorem, show that lim x->0 x^4(1-cosx)=0

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Hi, Maria,

WARNING -- USE COURIER FONT TO VIEW THIS.

When you use the 'sandwich theorem' you want three functions:

g(x) = top slice of bread
f(x) = peanut butter
h(x) = bottom slice of bread

and the peanut butter always stays inside the bread while you bite down so hard on the sandwich that the slices of bread come together.  So pick your functions so that:

1.  f(x) is your actual function, and
2.  h(x) <= f(x) <= g(x), and
3.  Both  g(x) and h(x) approach the same number at the limit.

What do you know about these functions?

Cos(x) has an amplitude of 1, which means it never goes above 1 or below -1.  Formally you write:

-1 <= cos x <= 1

-1 <= - cos x <= 1  << multiply by -1 and turn them around.
+1   +1         +1   << add 1 to each part
--------------------
0 <=  1 - cos x <= 2

Now construct our functions.  Multiply every part by  x^4:

0x^4 <=  x^4(1 - cos x) <= 2x^4

0    <=  x^4(1 - cos x) <= 2x^4
++++     ++++++++++++++    ++++
Your         Your          Your
h(x) <=       f(x)      <= g(x)

That's it.  Now just note that:

lim[x->0] h(x) = 0, because it is a constant.
lim[x->0] g(x) = 0, because it is a polynomial and g(0) = 0.

And now your sandwich theorem just Rolle's in.  Sorry, I mean it rolls in.