Two limit theorems

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 $$\lim_{x\rightarrow 3} (x^2-4),$$ we simply plug $3$ into $x^2-4$, getting 5. Another example: $$\lim_{x\rightarrow 4}\frac{x-2}{x+2}=\frac{4-2}{4+2}=\frac{1}{3}.$$

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 $(x^2-1)/(x-1) = x+1$ whenever $x \ne 1$, so $$\lim_{x \to 1} \frac{x^2-1}{x-1} = \lim_{x \to 1} (x+1) = 1+1 = 2.$$