Limit Laws#

Limit laws are a set of rules that tells us how to handle taking limits.

All limit laws need \(f(x)\) and \(g(x)\) to be defined for all values of \(x\) where \(x \ne a\) on an open interval.

We then say that:

\[ \begin{align}\begin{aligned}\lim_{x \to a} f(x) = L(x)\\\lim_{x \to a} g(x) = M(x)\end{aligned}\end{align} \]

With all the laws, try to think why it makes sense.

Addition#

The limit of sum of two functions is the sum of the limit of each function.

\[\lim_{x \to a}(L+M) = P\]

\[\lim_{x \to a} f(x) + \lim_{x \to a} g(x) = \lim_{x \to a} [f(x) + g(x)]\]

The limit of the difference of two functions is the difference of the limit of each function.

\[\lim_{x \to a}(L-M) = P\]

\[\lim_{x \to a} f(x) - \lim_{x \to a} g(x) = \lim_{x \to a} [f(x) - g(x)]\]

The limit of the multiplication of two functions is the multiplication of the limit of each function.

\[\lim_{x \to a}(L \cdot M) = P\]

\[\lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x) = \lim_{x \to a} [f(x) \cdot g(x)]\]

The limit of the division of two functions is the division of the limit of each function. Where \(g(x) \ne 0\)

\[\lim_{x \to a}{({L \over M})} = P\]

\[{{\lim_{x \to a} f(x)} \over { \lim_{x \to a} g(x)}} = \lim_{x \to a} [{f(x) \over g(x)}]\]