Math 201
Activity 2
Section 3
Spring 2013
sin x
x−→0 x
Squeeze Theorem and lim
y
6
M N
θ
O
P
Q
-
x
Assume that OM = OQ = 1
d
The arc M
Q is part of the circle with radius 1, centered at O.
\
1. Considering the segment M P and the arc M
OQ, prove that
0 ≤ sin θ < θ.
2. Use the Squeeze Theorem to prove that
lim sin θ = 0.
θ−→0+
3. In words, sketch a possible proof for
lim sin θ = 0.
θ−→0−
4. Now prove that
lim cos θ = 1.
θ−→0
[Hint: Find an identity involving sin θ and cos θ, solve for cos θ, and
use (3).]
5. Show that the area of the triangle 4ON Q equals
1
2
tan θ.
6. Considering (1), (5), and the area of the circular sector OM Q, prove
that
sin θ < θ < tan θ.
7. Thus show that
1<
θ
1
<
.
sin θ
cos θ
8. Hence
cos θ <
sin θ
< 1.
θ
9. Now prove that
sin θ
= 1.
θ−→0
θ
10. In words, sketch a possible proof for
lim+
lim−
θ−→0
sin θ
= 1.
θ