Some functions are full of holes when you graph them. What are holes, you might ask? Well, they’re not the kind that will cause a leak. Knowing what holes are, and how they relate to limits can be problematic, but the basic idea is very intuitive and helpful.
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This video finally gave me evidence of why certain graphs don't have tangent lines. The part of the video that gave me the "ahah" moment was the part where the first graph was made and explained. That graph gave me that moment because it explained why absolute value graphs don't have tangent lines at their vertex. Even though it has nothing to do with the topic the graph finally gave me evidence that the rule that I have been spoon fed throughout highschool has some sort of logical basis.
I thought that this pencast was a good review for the upcoming test for a few reasons. When I saw the title of it, it immediately reminded me of the recent topic of continuous functions that we just finished covering. The explanation of the function shows why some graphs have holes and are therefore not continuous at that point. It was also a good review of derivates/slopes because it shows why, even when h approaches 0 the 0 can not simply be substituted in for the denominator of the slope because this would make the function undefined. I thought he also made a good point of looking back at your work. Even after he factored out the numerator and cancelled out the denominator, the denominator still had much significance in the graph and the original function is key to the overall problem.
This video was extremely helpful. I always knew that you cancelled out same terms if they are in the numerator and denominator but no one ever really explained why. Now I know that it is because when you divide them it will always be one which will not change the outcome of the rest of the equation. I also liked this video because it helped me understand how to graph a function with holes. I was never really taught this either. I really liked how this explained it, it was simple and easy to follow!
I found this pencast helpful in solidifying why the graph of a line may have a hole in it (denominator=0). I've gone through the calculations many times and understood that this is what I was doing, but I don't think (or can't remember when) I've ever realized that the reason the function has a hole in it is because the function is undefined (denominator=0) at that point. This also ties back into continuous and continuable functions, piecewise functions, etc. – "ah-ha…."