Our First Limit
The code from the previous activity has created a point whose x coordinate is
h, and whose y coordinate is the slope of g. The point has been set up to leave a "breadcrumb trail" as we adjust h. Try it! Adjust h with its slider. There are two key things to observe here:
- There is a definitive trend in the slope of the secant line towards 4
- When
his set to 0 there is no secant line and so no slope
g when h tends to 0. As we can see, the limit of the slope of g, as h tends to 0, is 4.
In the next activity we'll see the standard algebraic calculation that verifies that this limit is equal to 4.Before we move on, I want to acknowledge a pretty reasonable question you might have: why bother with this "limit" thing at all, why not just set
h to 0 outright?
The answer is simply that the secant lines disappear when h becomes 0, so if we want to study the growth of f at A, all we can do is detect the trend in the slope of the secant lines as h tends to 0. It's clear from the above applet that the trend is towards the number 4.