By definition, we deal with

Lim_x->x0 f(x) 

Here, we must assume that f is defined in some neighborhood of x₀ except on x₀ itself

and from here that in the process of taking the limit, we have to assume that x approaches to x₀ in

any possible way but it is never equal to x₀

Example: Lim_x->5 {(x² - 25)/(x - 5)}

Here, in this example, we always have x ≠ x₀ = 5 during the limit process,

we can algebraically cancel the whole process

Lim_x->5 (x² - 25)/(x - 5) = Lim_x->5 {(x - 5)*(x + 5)}/(x - 5) = Lim_x->5 (x + 5) = 10

The above process shows that the original function behaves exactly as the sraight line y = x + 5

The original function is not defined at x = 5 is immaterial.

OR

Let you are given a rational function f : R{5} -> R, which is continuous everywhere in its domain.

Now, we want to find the limit of that function at 5

One way to do that is to construct a continuous extension of that function, g : R -> R such that

g(x) = f(x) whenever x is in the domain of f.

Then Lim_x->5 f(x) = Lim_x->5 g(x) = g(5)

In this case factoring and cancelling accomplishes that objectives.