Limit Laws
Limit laws allow us to compute limits by breaking down complex
expressions into simple pieces, and then evaluating the limit one
piece at a time. These laws are really theorems that have been
proven, based on the technical definition of the limit.
Limit Laws
Suppose that $\displaystyle\lim_{x \to a} f(x)$ and
$\displaystyle\lim_{x \to a} g(x)$ exist, and that $c$ is a
constant. Then:
- The limit of a sum is the sum of the limits: $$\lim_{x
\to a} \left(f(x)+g(x)\right) = \left(\lim_{x \to a}
f(x)\right) + \left(\lim_{x \to a} g(x)\right).$$
- The limit of a difference is the difference of the
limits: $$\lim_{x \to a} \left(f(x)-g(x)\right) =
\left(\lim_{x \to a} f(x)\right) - \left(\lim_{x \to a}
g(x)\right).$$
- The limit of a multiple is a multiple of the limit:
$$\lim_{c \to a} c \cdot f(x) = c \cdot \lim_{x \to a}
f(x).$$
- The limit of a product is the product of the limits:
$$\lim_{x \to a} \left(f(x)\cdot g(x)\right) =
\left(\lim_{x \to a} f(x)\right) \cdot \left(\lim_{x \to
a} g(x)\right).$$
- The limit of a quotient is the quotient of the limits
as long as you are not dividing by zero: $$\lim_{x \to
a} \frac{f(x)}{g(x)} = \frac{\displaystyle\lim_{x \to a}
f(x)}{\displaystyle\lim_{x \to a} g(x)}, \;\text{ if }\;
\lim_{x \to a} g(x) \ne 0.$$
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Notice that the same rules apply to limits as $x \to a^+$ or $x
\to a^-$.