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Easy: Polynomials
Polynomials are continuous everywhere, and rational functions are continuous wherever the denominator isn't zero. At such a point the limit is just the function's value: substitute x = a and simplify. Check the denominator first; if it is zero, substitution doesn't work and another method is needed.
Evaluate each limit.
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Whenever the function is continuous at the point, which includes every polynomial and any rational function whose denominator isn't 0 there.
Then the limit doesn't exist as a finite number; the function grows without bound near that point, often toward a vertical asymptote. If it gives 0/0, factor first.
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