Why must polynomial division be done prior to taking the limit? Suppose I wish to evaluate the following,
$$\mathop {\lim }\limits_{x \to 2} \left( {{{{x^2} - 4} \over {x - 2}}} \right)$$
If I just substitute two into $x$, it can't be done because the answer would be undefined (division by zero).
But, if I complete the polynomial division, that I hate to do because I'm all thumbs at it,
$$\mathop {\lim }\limits_{x \to 2} \left( {{{{x^2} - 4} \over {x - 2}}} \right) = \mathop {\lim }\limits_{x \to 2} \left( {x + 2} \right) = 4$$
Please tell me what's going on here?

Do not think of it as algebraic simplification. Think of the problem first. The original expression is a discontinuous function thus we are not able to solve for it. So we need to find an equivalent expression which is not discontinuous to make it easy for us to solve. Here simple algebraic operation gives us another function which is equivalent in the range we care about (between 0 and 2) and is continuous at 2, so we can solve for it. 
In other cases you may have to use trigonometric, logarithmic, or other  operations to find an expression that can be solved.
