This Mathwright Web activity uses zooming in to help students understand the geometric meaning of derivatives and limits. In the case of a limit, the idea is to consider the graph of a function over a punctured interval, and to repeatedly zoom in on the missing point. If the graph appears to be a curve with one missing point at all scales, then the limit exists, and equals the y value of the missing point. For the derivative, we again zoom in on a curve at a point. This time the object is to see if the curve appears to become a straight line under sufficient magnification. If it does, then the slope of that line is the derivative. Students work with the zooming in process using visual and numerical tools that emphasize these conceptual images.

There are two activity pages. The first screen (for limits) is shown below.

Students define the function to be studied, the point at which the limit will be determined, and the step size for computing sample function values (all shown in blue). Clicking a button causes the values of the function at 7 points on either side of the target x value to be printed numerically and plotted on a graph. One sided limits from either side can also be studied.

The second screen (for derivatives) is shown below.

Again, the student can enter the function, and the value of x at which a derivative will be determined, as well as the dilation factor by which the graph will be enlarged at each zoom operation. The graph of the function is plotted in blue, and a line is shown in green for comparison. The slope of the green line is controlled by a slider bar. Students are instructed to manipulate the slope until the green line agrees with the blue curve.