Limit of sin x/x as x Tends to Zero

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

x0

sinx

x

=1

.

This is the only limit needed to get the derivatives of all the trigonometric functions and their inverses.