Expert: Alon Mandes - 9/7/2008

Question
Hi !

I have another question, please.


My C.O.W. Tutorial  treats 1 and 2 below as two different situations.



1.Find the limit of a function at a point ,  for ex  
Lim of f(x) =(x^3+1)/2x at the point  x=2

vs

2. Find limx->2 (x^3+1)/2x.
 In this case, the CO.W. says that it is a question of finding how a function behaves near specific input values.


I do not understand the difference between 1 and 2  above,  because if we solve drill 1. by making f(x)= (2+h) and then  we do the substitution, we get lim= 9/4, which, if I am not mistaken,  is exactly the  same solution we get for  drill 2. by substituting  2 for x, as the tutorial teaches in its theory part.

Could you please shed some light  on the difference between the two expressions?


Thank you

Franco Vivona  

Answer
Hello Franco,
Indeed, what you said was true, you will get the same result either
by plugging the value 2 , or calculating (h+2). But what you must
realize that the function f(x) has no singularity at point x=2 !!
It's well defined there & continues. The difference between the two
treats you mentioned above, gets more clear when the point causes
the function singularity, like x=0 in our case, or x=3 for
f(x)=(x^2-9)/(x-3), or Ln(x-4) when x->4. In these cases we can't
just plug the value, but we have to approach the desired value, thus revealing the behavior of the function in the neighborhood.

Alon.