The standard geometric derivation for derivative of sinx depends on lim sinx/x, which in turn is based on the squeeze theorem. This broadly based and accepted misconception is the reason for this post.

Aside: You can’t define derivative of sinx from the series because you need derivative to get the series (OK, you can define sinx by the series, but then you have to show that it’s the same as the geometric definition, unless you want to abandon all geometrical meaning, in which case you have to derive trigonometric identities from series.)

The squeeze theorem: If

1) g(x) ≤ f(x) ≤ h(x) and lim g(x) = lim h(x) = L, Then Lim f(x) = L

From the geometric definition you get:

2) cosx < sinx/x < 1

It is then pretty universally concluded (Thomas Calculus, Wicki, google, etc, etc etc) that lim sinx/x = 1 by theSqueeze Theoremwhich is clearly incorrect since the conditions of the squeeze theorem are not satisfied.

The correct conclusion is that lim sinx/x = 1 by definition of limit, since by 2) sinx/x can be made arbitrarily close to 1.

In an effort to satisfy the requirements of the squeeze theorem many authors (I don’t know why they are so hung up on this) reach the following from the geometry: (google sinx/x and squeeze theorem)

cosx ≤ sinx/x ≤ 1

Which is obviously incorrect from the geometry.