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.

Limit of a Constant

\[\lim_{x\to a} c = c \]

The Sum & Difference Properties

\[\lim_{x\to a} [f(x) \pm g(x)] = \lim_{x\to a} f(x) \pm \lim_{x\to a} g(x)=L \pm M\]

The Constant Multiple Property

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

Generalized Product Property

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

Quotient Property

\[\lim_{x\to a} \dfrac{f(x)}{g(x)} = \dfrac{\lim\limits_{x\to a} f(x)}{\lim\limits_{x\to a} g(x)}=\dfrac{L}{M} ~~~~ \text {if $M\neq 0$}\]

The Power Property

\[\lim_{x\to a} [f(x)]^c = \left[\lim_{x\to a} f(x)\right]^c = L^c ~~~~ \text {if $L^c$ is defined}\]

Polynomial and Rational Functions

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

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