We can do arithmetic on limits in much the same way we do arithmetic on variables.

Here are the rules of limit arithmetic. They are mostly just common sense:

The limit of a sum is the sum of the limits.

$$ \lim [f(x) + g(x)] = \lim f(x) + \lim g(x) $$

The limit of a difference is the difference of the limits.

$$ \lim [f(x) - g(x)] = \lim f(x) - \lim g(x) $$

The limit of a product is the product of the limits.

$$ \lim [f(x) g(x)] = \lim f(x)  \lim g(x) $$

The limit of a quotient is the quotient of the limits., but only if the denominator is not zero.

$$ \lim {f(x) \over g(x)} = {\lim f(x) \over \lim g(x)} $$

The limit of the nth root is the nth root of the limits, but not for even roots of negatives.

$$ \lim \sqrt[n]{f(x)} = \sqrt[n]{ \lim f(x)  } $$

In applying these rules, we get some useful results that bear mentioning:

$$ \lim [f(x)]^n = [\lim f(x)]^n  $$

so that

$$ \lim_{x\rightarrow a} {x^n} = [\lim_{x\rightarrow a}{ x }] ^n = a^n  $$

it also follows from the rules that a constant factor can be moved through a limit sign.

$$ \lim k f(x)  =  \lim k \lim f(x)  =  k \lim f(x)  $$