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

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.)
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 . . .
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CalculusQuest^{TM}
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All rights reserved1996
William A. Bogley
Robby Robson