Scribe Post 9/18

Most of the class was spent on the quiz and IW#7. For this reason, Mr. O'Brien is scribing today...

We began by taking a look at question 5(a) from IW#7:

We graphed the numerator and denominator separately in GeoGebra:

We used a slider to explore the limit as x approached 0, 1, 2, and 3. We found the limits as x approached 0, 2, and 3 graphically by reading off the y values from the graph. These limits were 1/4, 0, and 1, respectively.

To find the limit as x approached 1, we zoomed in on the graph until we reached local linearity:

We then counted squares to find slopes and used the point slope form of a line to find the equation of each:

As x approaches 1, the numerator and denominator functions each get closer to their linearization. So, the limit can be rewritten as the limit of the two ratio of the two lines:

This generalization of finding the limit of an indeterminate form as the ratio of slopes is called l'Hopital's Rule.