Previously we used calculators and graphs to guess the values of limits, but we saw that such methods don’t always lead to the correct answers. Now we will use the following properties of limits, called the Limit Laws, to calculate limits.
Suppose that $c$ is a constant, $n$ is a positive integer, and the limits exist:
$$ \lim_{x \rarr a}f(x) \ \ \ \ \ \ \ \ \ \ \text{ and }\ \ \ \ \ \ \ \ \ \ \lim_{x \rarr a}g(x) $$
The limit of a sum is the sum of the limits.
$$ \lim_{x \rarr a}\left[ f(x)+g(x) \right] = \lim_{x \rarr a}f(x)+\lim_{x \rarr a}g(x) $$
The limit of a difference is the difference of the limits.
$$ \lim_{x \rarr a}\left[ f(x)-g(x) \right] = \lim_{x \rarr a}f(x)-\lim_{x \rarr a}g(x) $$
The limit of a product is the product of the limits.
$$ \lim_{x \rarr a}\left[ f(x) g(x) \right] = \lim_{x \rarr a}f(x)\cdot \lim_{x \rarr a}g(x) $$
The limit of a quotient is the quotient of the limits (provided that the limit of the denominator is not $0$).
$$ \lim_{x \rarr a}\left[ \frac{f(x)}{g(x)} \right] = \frac{\lim_{x \rarr a}f(x)}{\lim_{x \rarr a}g(x)} \ \ \ \ \text{if } \lim_{x \rarr a}g(x)\ne0 $$
The limit of a constant times a function is the constant times the limit of the function.
$$ \lim_{x \rarr a}\left[ cf(x) \right] = c\lim_{x \rarr a}f(x) $$
$$ \lim_{x \rarr a}\left[ f(x) \right]^n = \left[\lim_{x \rarr a}f(x)\right]^n $$
$$ \lim_{x \rarr a}\sqrt[n]{ f(x) } =\sqrt[n]{ \lim_{x \rarr a}f(x) } $$