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Taking two-sided limits using advanced algebra. I know how to solve it, but why does it work?

Hello, sorry if this has been asked before.

I'm studying calculus on Khan Academy and one of the required skills involves problems such as this:

lim x->1 (x^3 - 1) / (x^2 - 1)

Which is apparently unsolvable because you can't divide by zero, unless you play around with algebra and make it solvable:

(x - 1) (x^2 + x + 1) / (x + 1) (x - 1)

(x^2 + x + 1) / (x + 1)

And your limit is (1^2 + 1 + 1) / (1 + 1) = 3 / 2

Ok, fine. I know what to do. 

But *why*? How does a division with an undefined result magically become solvable because I played around with the variables? It doesn't make sense. Things tended to wrap around more tightly in HS algebra. What exactly is going on that solves the problem? Is there a way to explain to a non-mathematician?

Thanks in advance!
gpt
The limit does not include the value it is approaching. The reason why it works is because you changed the first function to form a second function that is equivalent at all values except for x = 1. The first function is the same as the 2nd, except there is a hole in the graph of the first function at x = 1. Because the limit does not include the value it is approaching, the value of each function at x = 1 is irrelevant; the values approaching 1 is what matters.