0/0 in limit evaluation When evaluating limits we often come across the form 0/0 and we cant get a limiting value. As stated in many textbooks and online resources, the method to still get a limiting value as the input approaches a specific value is to first factorize or simplify the function to cancel out the factors/variables responsible for the form 0/0.
My question is doesn't it change the function altogether if we simplify it?
Because if we had used the original function it would not have been defined at that particular input? Can anyone provide the intuition or concept behind this?

Let examine an example;
$$\lim_{x\to1} (\frac{x-1}{x^2 -1}) =\lim_{x\to1} \frac{x-1}{(x-1)(x+1)} = \frac{1}{2}$$
Since x is not 1 but around 1, we can do the simplification.
if we didn't do this, the function wouldn't be defined at 1 but it doesn't matter because we are not interested in the value of the function.
And it doesn't chance the function in matter value except at the point the function is undefined
