Transcript Calculus Lab: Limits and the Squeeze Theorem

Calculus Lab: Limits and the Squeeze Theorem
Finding limits at undefined points is easy if the function can be analytically transformed into another
x2 1
 lim x  1  2
function that agrees with its behavior in all but one point. For example, lim
x 1 x  1
x 1
sin x
lim
x

0
But other limits such as
x (which is known as one of the special trig limits) cannot be found
this way but do exist. Another method to find limits analytically is known as the Squeeze Theorem.
Step One: Copy the Squeeze Theorem from a calculus book and draw a sketch that could be used to
illustrate the Squeeze Theorem.
An Easy Example: Use g(x) = - | x | and h(x) =| x | to find the limit of f(x)=| x sin x | as x approaches 0
(which, by the way, could have been solved by direct substitution.)
Complete The Table:
x
h(x)
f(x)
g(x)
Sketch The Graphs:
0
A Harder Example: Let’s find functions g(x) and h(x) to Squeeze the limit lim sin x
x 0
x
We are going to use area of three regions on the unit circle.
y
C
B
Find a trigonometric expression to represent the areas of regions
named below and insert inequality symbols that make the statement true.
Manipulate analytically. Then apply the limits
θ
O
1
A
Triangle AOB
Sector AOC
Triangle AOC
A Harder Example: Let’s find functions g(x) and h(x) to Squeeze the limit lim sin x
x 0
x
We are going to use area of three regions on the unit circle.
y
C
B
Find a trigonometric expression to represent the areas of regions
named below and insert inequality symbols that make the statement true.
Manipulate analytically. Then apply the limits
θ
O
1
A
Triangle AOB
Sector AOC
Triangle AOC
Finally: Rewrite the Squeeze Theorem here in your own words and graph an example of your own
Finally: Rewrite the Squeeze Theorem here in your own words.
The Other Special Trig Limit
Investigation:
1  cos x
Estimate the lim
by graphing the functions on a graphing calculator and by
x 0
x
creating a table. Show your graph below, remember to label the window.
xx
0
f(x)
Proof:
Manipulate the Pythagorean Identity
sin 2 x  cos 2 x  1
into two equivalent functions that can be used to find
1 cos x
)
x
(hint one of the functions will have to be
lim
x 0
1  cos x
x
y
C
B
θ
O
A
1
y
C
B
θ
O
1
y
C
A
B
θ
O
1
A