Two limit theorems
Theorem:
If $f$ is a polynomial or a rational function, and $a$ is in
the domain of $f$, then $$\lim _{x\to a} f(x)=f(a).$$ 
This theorem is true by virtue of the earlier limit laws. By
applying the product rule, we can get $\displaystyle\lim_{x \to a}
x^n = a^n$. Combining this with our rule for multiples and sums
gives the theorem for polynomials. Combining that with our rule
for quotients gives the theorem for rational functions.
In practice, the theorem says that whenever $f$ is a polynomial
or rational function, we can evaluate $f$ at $a$, and if this
value exists, it is the limit as $x$ approaches $a$.
For example, if we wish to evaluate $$\lim_{x\rightarrow 3}
(x^24),$$ we simply plug $3$ into $x^24$, getting 5. Another
example: $$\lim_{x\rightarrow
4}\frac{x2}{x+2}=\frac{42}{4+2}=\frac{1}{3}.$$ The conclusion: if you plug
$a$ into $f$ and there are no issues (dividing by zero, negative
under a square root, etc.) most likely that is the limit value.
Theorem:
If $f(x) = g(x)$ whenever $x \ne a$, then
$\displaystyle\lim_{x \to a} f(x) = \lim_{x \to a} g(x)$. 
When $x$ is close to $a$ (but not equal to $a$), $g(x)=f(x)$, so
$g(x)$ is close to whatever $f(x)$ is close to. This seems like a
trivial result, but it is very useful for evaluating limits of
ratios where both the numerator and denominator go to zero. For
instance $(x^21)/(x1) = x+1$ whenever $x \ne 1$, so $$\lim_{x
\to 1} \frac{x^21}{x1} = \lim_{x \to 1} (x+1) = 1+1 = 2.$$
The conclusion: If you
can factor the top and bottom and cancel a factor from both, and
the factor is zero at $x=a$, then the original function and the
function you did more work on are the same except at $x=a$, so
their limits are the same as $x\to a$. To restate slightly
differently, notice that this limit was of the form $\tfrac00$, so
we needed to do more work.
The work we did was getting rid of common factors in the top and
bottom, which we could do because of the second theorem
above.
