x > sin(x)
for all x > 0. It can be easily seen if you think about the circle / triangle interpretation or think about the series expansion of sin(x)
x - x^3/3! + ...
Not sure about the x->0 part of your sketch if its
(x + sin(x))/(x-sin(x))
It will be something like ~ 1/x^2 and hence a vertical asymptote at x=0.
For "large" x, it will be something like 1+sin(x)/x, so the mexican hat which is sin(x) where the magnitude goes down like 1/x.