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WOLFRAM|DEMONSTRATIONS PROJECT

Limit of sin x/x as x Tends to Zero

​
0.250
0.262
0.289
Express
ΔOPM>sector(OPQ)>ΔOPQ
as
1
2
tanx≥
1
2
x≥
1
2
sinx
. Divide by
1
2
sinx
and take reciprocals to get
cosx≤
sinx
x
≤1
. Since
cos0=1
, by the squeeze theorem,
lim
x0
sinx
x
=1
.
This is the only limit needed to get the derivatives of all the trigonometric functions and their inverses.
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