### Limits Need Not Exist

We have a mystery function s(u) and want you to guess

 lim s(u) u-->0

Plug in various values of u on either side of 0 and then guess the limit.

 u =           s(u) =

We have asked you to find the limit as u approaches 0 of the function

s(u) = u/|u|.

If u>0, then |u| = u, so s(u) = 1. If u<0 then |u| = -u, so s(u)=-1. The function s(u) is not defined for u=0. Its definition as a piecewise function and its graph are given below:

 { 1 if u > 0, s(u) = undefined if u = 0, -1 if u < 0.

But . . . what value could

 lim s(u) u-->0

possibly have?

If L is the limiting value, then |s(u) - L| is supposed to get arbitrarily small for points u sufficiently close to zero. But there are points arbitrarily close to zero (to the right of zero) where s(u) = 1, and other points arbitrarily close to zero (to the left of zero) where s(u) = -1. And no number L can simultaneously be arbitrarily close to both -1 and 1. In fact, there is no number that can be closer than 1 unit away from both 1 and -1.

Given a function f(x) and an objective a, if there is no value L such that

 lim f(x) = L, x --> a

then we say that the limit as x approaches a of f(x) DOES NOT EXIST. This will be indicated by writing

 lim f(x) = DNE. x --> a

Here is another example of a limit which does not exist. The function is a variation on a familiar function - can you figure out what it is? The limit we are interested in is the

 lim g(x) x-->2

(Enter values of x and press the "check" button to see the values of g(x): press the "explanation" button to find out which function you are examining.)

 x =           g(x) =

Now that we know limits need not exist, must we prepare to meet our fates at the hand of Smith? Will Theseus save us? It's time to see . . .

/Stage3/Lesson/DNE.html