What does limit of a derivative imply? If the limit of a derivative of a function when x tending to infinity exists, it is zero. ( Considering limit of the function at x tending to infinity is finite.)
I wonder what this limit implies, physically or graphically.

As Daniel pointed out, your assertion is incorrect. If a function approaches some value L (presumably a horizontal asymptote) when $x$ goes to infinity, then the derivative does not have to go to zero at all. Under "high school" circumstances, it usually will (most of those functions behave kind of "nice"?), however, consider the following counter example: $y=\frac{sinx^2}{x}$. It is clear that for large $x$, the $y$ goes to zero. The derivative of this function is $y'=\frac{2x^2cosx^2-sinx^2}{x^2}$ which simplifies to $2cosx^2-\frac{sinx^2}{x^2}$. Is it clear that the derivative will never go to zero for larger $x$?
