The code from the previous activity has created a point whose x coordinate is [code]h[/code], and whose y coordinate is the slope of [code]g[/code]. The point has been set up to leave a "breadcrumb trail" as we adjust [code]h[/code]. Try it! Adjust [code]h[/code] with its slider. There are two key things to observe here:[br][br][list][*]There is a definitive trend in the slope of the secant line towards 4[/*][*]When [code]h[/code] is set to 0 there is no secant line and so no slope[/*][/list][br]If we want to study the trend of the slope of g and conclude that it terminates at a fixed value, we need to use the idea of a [b]limit[/b], a mathematical tool for studying patterns that never terminate, such as this one.[br][br]For instance, in this situation we can study the [b]limit[/b] of the slope of [code]g[/code] when [code]h[/code] tends to 0. As we can see, the limit of the slope of [code]g,[/code] as [code]h[/code] tends to 0, is 4. [br][br]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 [code]h[/code] to 0 outright? [br][br]The answer is simply that the secant lines disappear when [code]h[/code] becomes 0, so if we want to study the growth of [code]f[/code] at [code]A[/code], all we can do is detect the trend in the slope of the secant lines as [code]h[/code] tends to 0. It's clear from the above applet that the trend is towards the number 4.