Properties of Finite Limits

Properties

Le \(c\) be any real number and suppose \(\displaystyle \lim_{x\to a} f(x) = L\) and \(\displaystyle \lim_{x\to a} g(x) = M\), where both \(L\) and \(M\) are finite values.

\[\begin{align*} &\text{Limit of a Constant} && \lim_{x\to a} c = c \\ \\ &\text{Sum & Difference Properties} && \lim_{x\to a} [f(x) \pm g(x)] = \lim_{x\to a} f(x) \pm \lim_{x\to a} g(x)=L \pm M \\ \\ &\text{Constant Multiple Property} && \lim_{x\to a} cf(x) = c\lim_{x\to a} f(x) = cL\\ \\ &\text{Generalized Product Property} && \lim_{x\to a} [f(x)g(x)] = [\lim_{x\to a} f(x)][\lim_{x\to a} g(x)]=LM \\ \\ &\text{Quotient Property} && \lim_{x\to a} \dfrac{f(x)}{g(x)} = \dfrac{\lim\limits_{x\to a} f(x)}{\lim\limits_{x\to a} g(x)}=\dfrac{L}{M} ~~~~ \text {if $M\neq 0$} \\ \\ &\text{Power Property} && \lim_{x\to a} [f(x)]^c = \left[\lim_{x\to a} f(x)\right]^c = L^c ~~~~ \text {if $L^c$ is defined} \end{align*}\]

The Limit of a Polynomial or Rational Function

If \(f(x)\) is a polynomial or rational function and \(a\) is in the domain of \(f(x)\), then

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