Why we need limits to evaluate the "slope" of a changing funciton.

In order to find the instantaneous rate of change at a single point along a curved (changing) function, we find the slope of a short line segment (called the secant line) whose "run" () is very short. This line segment becomes a good approximation of the slope of the curve, the closer gets to zero. However, if is allowed to actually equal zero, two bad things happen. First, also becomes zero, because in zero time we move zero distance. But perhaps worse, we find ourselves dividing .
To solve this dilemma, we employ a tool called a limit. The idea of the limit is to get as close to zero with out actually making it zero, and seeing what happens to . As long as , the ratio approximates the slope at this point. We can make as small as we like. The smaller it is, the closer we get to the exact answer. Using limits allows us to actually get the exact answer without worrying about dividing by .
In the left pane of the app, the red graph represents the position of an object falling off a tall cliff. The slope of this position graph gives us the velocity of the object. Here you'll see a blue dot and a purple dot on the -axis. The purple dot tells us the point in time at which we are trying to find the slope , and the blue dot lets us set to any value near the purple dot target. At the single point on the graph corresponding to the purple dot, a tangent line is drawn, whose slope is exactly equal to the instantaneous velocity at that time. By moving the blue dot toward the purple dot, the secant line slope (the approximation between two points with ) approaches the tangent line slope (the exact value at a single point).
What you'll notice is that if you move the blue dot exactly onto the purple dot, becomes zero and the secant line's slope is undefined. However, when the blue dot is very, very close to the purple dot, but not actually at the exact same place, , and the secant slope is defined.
These characteristics of the position function have powerful implications for the velocity function. The graph on the right shows the velocity calculated for a given "purple dot", for all possible values of the "blue dot". Notice that when the blue and purple dots coincide, the slope of the position (= velocity) is undefined, and we have a hole in the velocity graph just where we need a value. But visually, it looks clear that the value that "should" be there is the value given by the limit of the velocity on either side of the hole. For example, with the purple dot set to , the velocity "hole" is at a velocity value of ft/s. Although there is a hole at this value, the velocity graph seems to "pass through" the hole, and it seems intuitively reasonable that this really is the exact instantaneous velocity at .
Try setting the purple dots to various values of and study the results.