1-D Limits of Difference Quotients
Instructions
In this lesson we are going to learn algebraic techniques for evaluating limits. The main motivation for being able to do so is to evaluate the limit of a difference quotient expression (i.e., average rate of change) in order to obtain an instantaneous rate of change.
- The left part of the graph shows the motion of a moving point, including an arrow to represent the average and instantaneous rates of change.
- Use the slider tool for h to move the point Q around the point P.
- Use the respective checkboxes to show/hide the secant between P and Q, the tangent line at P, and the graph of the difference quotient function.
- When the Secant box is checked, a point will appear. This point takes the slope of the secant line and represents it as a y-coordinate. Adjusting the value of h shows how this slope is a function of h; this is the difference quotient function. Notice it is undefined when h=0, but what value does it seem to approach?
1-D Evaluating Limits Algebraically
This preview is about introducing how we will use limits of average rates of change to find instantaneous rates of change. Our goal in the lesson will be to actually evaluate limit expressions exactly without having to rely on estimation techniques with graphs and tables. In other words, we will take a purely algebraic approach using the difference quotient expression.
Almost all algebraic techniques for evaluating limits boil down to simplifying an expression to a point of "canceling" a common factor. If you look at the difference quotient function graph, you will notice that this function always has a discontinuity where h=0. If that discontinuity is removable (i.e., a hole), then canceling the common factor is essentially the algebraic way of filling in this hole so that we obtain a continuous extension for the difference quotient function. (If the discontinuity is not removable, we will talk later about how this means that the instantaneous rate of change does not exist there.)