Limits Need Not Exist

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

lim s(u)

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

u =     Submit Value      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:
Graph of s(u)=u/|u|

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

But . . . what value could

lim s(u)

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)

(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 =     Submit Value      g(x) =

     Submit Value

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 . . .



© CalculusQuestTM
Version 1
All rights reserved---1996
William A. Bogley
Robby Robson