Expert: Alon Mandes - 7/1/2010

Question
Pick a point c such that the limit of f(x) as x approaches c from the right and the left are equal, but the function is not defined at the point c and the function is not continuous at c.  
I've picked the function : f(x) = (x+2)(x-2)/x-2
I know that c=2 and x=2 is undefined, but I'm a bit confused as to the limits.Could you please explain this and show any work. Thank you very much.

Answer
Hello Cathy,
The function that you suggested is f(x)=(x²-c²)/(x-c) .
We can see that x=c is a singularity point for f(x) , because f(c) is from the form 0/0 &
it's not defined. BUT, the limit exists.

Lim  (x²-c²)/(x-c) =
x->c
Lim  (x+c)(x-c)/(x-c) =
x->c
Lim  (x+c) = c+c= 2c .
x->c

Further more, the side limits exist & equal :
Lim   f(x)   =   Lim  f(x)   =  Lim  f(x)  = 2c .
x->c-            x->c+          x->c  

If you try to sketch this function using a software program, you will discover that the graph
is totally continues . But you were told to give a function that is not continues at x=c .
Therefore , f(x) doesn't sasisfy the requirement.

Let's consider the function : f(x)=(x²-c²)/(x-c) for x≠c AND at x=c f has value of c .
In this case , we satisfy ALL the requirements . You can see now that
Lim  f(x) ≠ f(c) . So the function is not continues at x=c !!
x->c

ALon.