# NOTES 01.1 Limits & Continuity.pdf - Calculus Maximus Notes...

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Calculus MaximusNotes 1.1: Limits & ContinuityPage 1 of 11§1.1—Limits & ContinuityWhat do you see below?We are building the “House of Calculus,” one side at a time . . . and we need a solid FOUNDATION.
Calculus MaximusNotes 1.1: Limits & ContinuityPage 2 of 11Example 1:(Calculator) For2( )f xx(a) fill in the following chartx2.92.992.9993.13.013.001(fx(b) What do these values tell us aboutfin theneighborhoodof3x(c) Based on the chart above, what do you think is the value of(3f(d) What IS(3fExample 2:(Calculator) If323( )3xxg xx-=-,(a) fill in the following chartx2.92.992.9993.13.013.001(g x(b) What do these values tell us aboutgin theneighborhoodof3x(c) Based on the chart above, what do you think is the value of(3g(d) What IS(3g(e) Is there a still a way to mathematically communicate the result from the chart above?==???=???
Calculus MaximusNotes 1.1: Limits & ContinuityGoing where we can’t go . . .This interruption to the flow of the graph ofgin Example 2 is called aremovable point discontinuity, oraholein the graph ofg.Simply evaluating a function at a particular value is insufficient for understanding the behavior of sometypes of functions at that point, especially functions with discontinuities at those points.There is, therefore,a need to come up with another method that will circumvent the possibility of going directlytoa location,but ratherapproachingthat location from either side of it.This is the limit, and it has its own notation asyou will seeThe Limit is a Notion of MotionExample 3:Suppose you wanted to safely and smooooothly cross achasm in your car.What three things would you need?Would you be able to do so in the situation depicted in thephoto to the right?What is missing?If you were to build the bridge, would you know exactlywhere to build it to make crossing the chasm in a car acontinuously smooth undertaking?
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