If \(\displaystyle \lim_{x\to a} f(x) \) and \(\displaystyle \lim_{x \to a} g(x)\) exist, then we have the following properties:

1. \(\displaystyle \lim_{x\to a} k = k\), where \(k\) is a constant.
2. \(\displaystyle \lim_{x\to a} x = a\).
3. \(\displaystyle \lim_{x\to a} \left [ f(x)\pm g(x) \right ] = \lim_{x\to a} f(x) \pm \lim_{x\to a} g(x) \).
4. \(\displaystyle \lim_{x\to a} cf(x) = c \left[ \lim_{x\to a} f(x) \right]\).
5. \(\displaystyle \lim_{x\to a} \left [ f(x)g(x)\right ] = \left [ \lim_{x\to a} f(x) \right ] \left [ \lim_{x\to a} g(x) \right ]\).
6. \(\displaystyle \lim_{x\to a} \dfrac{f(x)}{g(x)} = \dfrac{\displaystyle\lim_{x\to a} f(x)}{\displaystyle\lim_{x\to a} g(x)}\), provided \(\displaystyle\lim_{x\to a} g(x) \neq 0\).
7. \(\displaystyle \lim_{x\to a} \left [ f(x) \right ]^{~r} = \left [ \lim_{x\to a} f(x) \right ]^r\), for most real numbers, \(r\).
   Note: If \(r\) is negative, then we are assuming \(\displaystyle \lim_{x\to a} f(x) \neq 0\). If \(r=\frac{p}{q}\) in lowest terms with \(q\) even, then we are assuming that \(\displaystyle \lim_{x\to a} f(x) \geq 0\).