Very quickly ... Say you have an expression like and you want to find f(2).
Now it's clear that plugging 2 into the expression won't help because you get 0/0 which is undefined.
But if you creep up on it from above, setting x = 3, x = 2.5, x = 2.1, x = 2.01, x = 2.001 ... and so on, you get a sort of picture of what you think the answer is also trying to creep up on, which (as you'll find out when you try it) is 4. That's approaching the limit from above, and can be written:
You can do the same thing by creeping up on it from below, setting, for example, x=1, x=1.5, x=1.9, x=1.99, etc., which can be written:
(as you'll find out when you try it out).
Now, if it just so happens that your expression works out as the the same whether you creep up on it from above OR below, then you don't need to write which direction you came to it from, and can write:
Okay, so how do we "know" it's 4? Because
but this ONLY HOLDS when because although most of the time, whatever algebraic manipulations you care to play with, you can NOT get to have a value when x = 2, ALL YOU CAN DO is say that it "f(x) tends to a limit of 4 when x tends towards 2."
Does this help?