Properties of Finite Limits

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.

(1) Limit of a Constant

For any real number \(c\),

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

(2) The Sum & Difference Properties

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

(3) The Constant Multiple Property

For any real number \(c\),

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

(4) Generalized Product Property

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

(5) Quotient Property

If \(M\neq 0\),

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

(6) The Power Property

For any real number \(r\),

\[\lim_{x\to a} [f(x)]^r = \left[\lim_{x\to a} f(x)\right]^r = L^r\]

if \(L^r\) is defined.

(7) Polynomial and Rational Functions

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

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