Simple Graphing

This notebook is to simply show how we can use "Plot" to analyze limits in calculus I. This is to supplement more rigorous arguments not replace them in most instances.

Problem 9 part a.)

Plot[(x+3)/(x-1), {x, 0,2}]

It should be clear from this picture what the limit of the function is as x⟶(SuperPlus[1]).

Problem 9 part b.)

Plot[ (2x-9)/(x-4)^2, { x,2,6}]

The limit of the function graphed above at x=4 should be obvious.

Problem 9 part d.)

Plot[Sec[x],{x,0,2}]

The limit of f(x) = sec(x) as x⟶ π/(SuperPlus[2]) is clearly indicated by the graph above.

Advice

You don't want to use Mathematica to calculate these limits, however it is a good idea to check your intuition with this tool. Remember, on the test you only have your mind and a scientific calculator at your disposal. Your principle tools are considering values very close to the limit point to frame a guess then if appropriate you'll use the limit laws and algebraic manipluation to reduce the limit so that the direct substitution property applies. Which method of analysis is appropriate depends on how we ask the question:

	a.) "use intuitive and/or numerical methods to deduce the value of the limit" this suggests 		you make table of values and/or think about graphs to make an educated guess.
		
	b.) "use algebra, limit laws and the direct substitution property to determine the value of 		the limit. Show how any indeterminancies are resolved if possible, or show how 		they are not" this instruction suggests you can use algebra and limit laws to cancel 		factors etc...this is probably what you remember from previous calculus courses if you 		had one.
	
	c.) "use the limit laws to calculate the following limit" here we have annoyingly removed   	      	the ever-so convenient "direct substitution property" that I worked slavishly to establish 		in the third week of lectures. The standard of proof here is more  legalistic than (b) but 		not nearly as serious as what follows below.
	
	d.) "use the ϵδ-definition of the limit to prove that f(x) ⟶ L as x ⟶ a" here you have to   	         	do scratchwork to work backwards to choose the δ (typically as a function of ϵ) then 		once that is done you start at the beginning and methodically show the definition of the 		limit is	satisfied for the given limit.