Epsilon-Delta Part II

I.  Quiz

II.  Homework  (not adjustment for problems 12 and 13 should state at x = 2 and  at x = 1)

III.  A proof of a limit that does not exist

Example:  Prove that the function

f(x) = x² - 1 for x < 2 and

x + 3 for x > 2 

does not have a limit at x = 2

Solution:  let epsilon = .5

then for any chosen delta, chose m =min(delta/2,.01) so that

f(2 - m) = (2 - m)² - 1 = 3 - 2m + m²  < 3.1 and

f(2 + m) = (2 + m) + 3 = 5 + m > 4.9.

Now for any L either

|3.1 - L| > .5 or |4.9 - L| > .5

hence the limit does not exist.

Exercise:  Prove that if f(x) = 3x - 5 for x < 1 and 2x + 2 for x > 1, then the limit does not exist.

IV.  Limits and Graphs

If f(x) is a function, then the limit as x approaches c is L if the y coordinates of the left hand side from x = c of the graph and the right hand side of the graph both approach L.  Many graphical examples will be given.  Graphically, we can find the limit by putting the function into a graphing calculator and checking to see if the left and right agree and the y coordinate is the limit.