Trying to understand limits fundamentally: Why is lim (x->2) (x^2 - 4)/(x - 2) = 4 if plugging in 2 gives 0/0?

Hey everyone, I'm starting out with calculus and really want to build an intuitive understanding of limits from the ground up rather than just mechanically applying algebraic tricks.

Take this standard limit:

lim (x → 2) (x² - 4) / (x - 2)

Algebraically, the steps are straightforward:

1. Factor the numerator: (x - 2)(x + 2)
2. Cancel out (x - 2) from top and bottom to get x + 2
3. Substitute x = 2 to arrive at 4

Here is where my conceptual confusion lies:

• If I plug in x = 2 directly at the start, the expression is 0/0, which is undefined.
• In normal algebra, dividing by zero is forbidden. If x = 2, then (x - 2) = 0, so shouldn't canceling (x - 2)/(x - 2) be dividing zero by zero?
• Why does the limit equal 4 when the function value at x = 2 doesn't even exist?

Could someone explain the fundamental philosophy of what a limit is doing here, both conceptually and graphically, and why this algebraic cancellation is valid?