Comment by enizor2

I do not understand this consideration: > By considering a triangle with hypotenuse 1 and a very small “opposite” side, it’s not hard to see geometrically that sin(x)≈x and cos(h)=x when x is small

I fail to see how you can "see" finer than sin(h) -> 0 & cos(h) -> 1

From the limit definitions you actually need :

* (1-cos(h)) / h -> 0

* sin(h)/h -> 1

(which correspond to the derivatives at 0).