Evaluating Limits: Question & Solution

[frozonecom]

Homework Statement

Question : lim_{x\rightarrow2}f(x)​

Homework Equations

The Attempt at a Solution

I am really very confused about this. I know limits are the "intended height of the function". So, what if the function is like this? I know for one that this is a function since it passed the vertical line test. What I'm confused about is how to identify its limit.

Should it be 3, or 5? By substitution(by evaluating the function itself), based from the graph, the value is 5.

But, is it possible to get another solution for that kind of function by using factoring or the conjugate method? Perhaps "3"? Or is it impossible to get different values for the limits by using different methods of evaluating limits?

So, does a limit exist in that kind of function?

I'm asking this because in our test, our teacher asked this kind of question. She said that it should be 3, because it is clearly "what the height intends to be". THIS WHAT CONFUSED ME.
Thanks in advance! :)

 

[HallsofIvy]

This is a crucial point in understanding limits. The definition of limit says
"\lim_{x\to a} f(x)= L if and only if, given \epsilon> 0 there exist \delta> 0 such that if 0< |x- a|< \delta, then |f(x)- L|< \epsilon"
The reason I show that is to emphasize the "0< " in that definition. What happens when x= a is irrelevant! In determining the limit the question is "what is f(x) close to if x is close to (but NOT equal to) a?"