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The steps to calculate the limits of the functions sin(x) and cos(x) as x approaches 0 using numerical, geometric, and algebraic methods. It also includes the use of fermat's definition of derivatives and the squeeze theorem. Taken from a university mathematics course, specifically math 1501.
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Derivative of
x
sin x
sin
lim
0
x
x
x
x
x
x
sin x
1
sin
lim
0
x
x
x
x
sin x
sin
lim
0
x
x
x
2
hOB r OB BC
(cos( ) 1 )
(cos 1 )
lim
2
0
h h
h
h
2 2
2 2
h h
(cos( ) 1 )
sin
lim
cos( ) 1
lim
2
0 0
h h
h
h
h
h h
cos( ) 1
lim
sin( )
lim( sin( )) lim
0 0 0
h h
h
h
h h h
x
dx
d x
cos
sin
f ( x )sin x
h
0
h
0
h
0
h
0
h h
0 0
0
h
h h
0 0
0
h
0
h
x x x
dx
df
sin 0 cos 1 cos
f ( x )sin x
x
dx
df
f ( x ) cos
