A nice treatment of limits like sin(x), x-> 0 can be found here:

https://ocw.mit.edu/courses/mathematics/18-01-single-variable-calculus-fall-2006/video-lectures/lecture-3-derivatives/

In particular, he illustrates sin(x) vs. the chord for small angles, which makes it obvious geometrically that the limit is the angle for small angles (since the circular arc and chord end up approaching each other).