The tangent line problem deals with how to determine the slope of a non-straight function at a specific point, . We can approximate that slope using the slope of a secant line through the point and a second point that we pick to be reasonably close to the first point.
In this applet you can choose any -value you like in the range . Then you can let vary from -1 to +1 using the slider. The value chooses how far away the second point is from our point of interest.
Notice that as approaches 0, from either the left or the right, the slope of the secant line gets closer to the desired slope of the tangent line. This is what happens as we consider .