I was reading Calculus by Michael Spivak, when I came across this problem given in the textbook..  

Compute the limit
$$\lim\_{x\to 2}{\frac{x^3-8}{x-2}}$$
Now, this is a pretty standard limit and I know there are a couple of ways to solve it..by factoring or by using L Hospital’s rule...but here..I want to solve the limit by factoring..

My question is, when we solve the limit by factoring..we make an assumption that $x$ is not equal to 2. Now, yea I know that $x$ is only approaching the value 2..and hence the assumption is justified...but then this is not what the definition says right? 

To me, the definition of a limit allows us to ignore the value 2 
,while verifying that a certain number is the limit of the function or not..by finding a $\delta$ given an $\epsilon$.

What I mean to say is..the word “approach” is quite subjective and to me it’s not very convincing that the limit will be the same even if we ignore the value 2...so I am looking for a rigorous line of reasoning entirely based on the definition of a limit which would then allow us to completely ignore the value of the function at $x=2$ even while COMPUTING the limit...not just while VERIFYING whether a certain number is the limit or not

PS: My question is a little “dry”, in the sense it asks for something which should be obvious..but I’m still confused..because the definition of the limit is something very fundamental..and any such assumption made while computing limits must follow from the definition..

EDIT: I have tried to explain my question in a better way [here](https://math.stackexchange.com/questions/3663812/how-can-we-ignore-the-limit-point-while-computing-the-limit-of-a-function)

Thanks for any answers!!