But wait a second, the breadcrumb trail doesn't [i]exactly[/i] match[code] f'(x)[/code]. Here's a zoomed in view to make this easier for you to see. Move [code]A[/code], and notice that the breadcrumb is systematically too high.

Don't worry. We haven't done anything wrong. We just haven't applied what we learned from [url=https://www.geogebra.org/m/x39ys4d7#material/zruqdnrq]our previous study of [b]Limits[/b][/url]. [br][br]The problem is, in order to exactly match [code]f(x)[/code] at [code]A[/code], we must use the [i]limit[/i] of the slope of [code]g[/code] as [code]h[/code] approaches 0. In this activity, [code]h[/code] is set to .01. We need the [i]limit[/i] of the slope secant lines as h approach 0. Any specific reading on [code]g[/code] isn't good enough. We need the line at "end of the trend". [br][br]Recall from [url=https://www.geogebra.org/m/x39ys4d7#material/wjqeqb8g]our first limit[/url] in the previous chapter, when [code]A=(1,3)[/code], the line at the end of the trend had slope 4. But this number will change when [code]A[/code] moves to other points on the graph of [code]f(x)[/code]. [br][br]How do we deal with this? Move forward to see one idea.