1. The problem is to understand the meaning of the expression $$\frac{\Delta y}{\Delta x} \to 0$$ and its relation to the derivative $$\frac{dy}{dx}$$. 2. The expression $$\frac{\Delta y}{\Delta x}$$ represents the average rate of change of a function over an interval $$\Delta x$$. 3. When $$\frac{\Delta y}{\Delta x} \to 0$$ as $$\Delta x \to 0$$, it means the average rate of change approaches zero. 4. The derivative $$\frac{dy}{dx}$$ is defined as the limit of the average rate of change as $$\Delta x$$ approaches zero: $$ \frac{dy}{dx} = \lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} $$ 5. Therefore, if $$\lim_{\Delta x \to 0} \frac{\Delta y}{\Delta x} = 0$$, then the derivative $$\frac{dy}{dx} = 0$$. 6. This means the function has a horizontal tangent line at that point, indicating no instantaneous change in $$y$$ with respect to $$x$$ at that point. 7. In summary, $$\frac{\Delta y}{\Delta x} \to 0$$ implies $$\frac{dy}{dx} = 0$$, which is the slope of the tangent line at the point of interest.