Lesson 3
SQUEEZE THEOREM
SQUEEZE THEOREM
LIMITS OF FUNCTIONS USING THE SQUEEZE PRINCIPLE
The Squeeze Principle is used on limit problems where the usual
algebraic methods (factoring, conjugation, algebraic manipulation,
etc.) are not effective. However, it requires that you be able to
``squeeze'' your problem in between two other ``simpler'' functions
whose limits are easily computable and equal. The use of the Squeeze
Principle requires accurate analysis, algebra skills, and careful use of
inequalities. The method of squeezing is used to prove that f(x)→L as
x→c by “trapping or squeezing” f between two functions, g and h,
whose limits as x→c are known with certainty to be L.
SQUEEZE PRINCIPLE :
Assume that functions f , g , and h satisfy g(x) £ f(x) £ h(x)
and
then
lim g(x) = L = lim h(x)
x ®a
x ®a
lim f(x) = L
x ®a
EXAMPLES:
Evaluate the following limits.
tan x
1. lim
x ®0
x
sin 3 x
3. lim
x ®0 sin 5 x
sin 2 x
2. lim
x ®0
x
2 - cos3x - cos4x
4. lim
x ®0
x