Doesn't solving limits by substituting values defeat the point? For example: you can solve a limit of a quadratic by just subbing in a x value. But how do we know limit as x approaches a is the same as f(a)?
I'll try to take a stab at what you're getting at :)
For limits, we are concerned what y-value the function is headed for. It doesn't matter if there is a hole or something at that point, because the limit never actually equals the value x is headed towards. It can get infinitely close, however.
For some limits, we don't have to do much work or simplification. So, if you can plug in the x-value without the denominator being equal to 0, go for it!
Also, if lim x-> a of f(x) exits and equals f(a) , we call the function continuous (Assuming f(a) exists as well)