Let \(a\) be any number.

1. For any number \(c\), \(\lim_{x \to a} c = c\)
2. \(\lim_{x \to a}x = a\)

Suppose \(\lim_{x \to a}f(x) = L\) and \(\lim_{x \to a}g(x) = M\). Then:

1. \(\lim_{x \to a}(f(x) + g(x)) = L + M\)
2. For any number \(c\), \(\lim_{x \to a}cf(x) = cL\)
3. \(\lim_{x \to a}(f(x) - g(x)) = L - M\)
4. \(\lim_{x \to a}(f(x) . g(x)) = L . M\)
5. If \(M \neq 0\) then \(\lim_{x \to a}(f(x) / g(x)) = L / M\)

If \(f\) is a polynomial, then for any number \(a\)

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

Suppose that \(d > 0\), and \(f_1\) and \(f_2\) are functions such that for all \(x\), if \(0 < |x-a| < d\) then \(f_1(x) = f_2(x)\). Then \(\lim_{x \to a} f_1(x) = \lim_{x \to a} f_2(x)\), where we interpret this equation to mean that either both limits are defined and they are equal, or both limits are undefined.

If \(\lim_{x \to a}(f(x)/g(x))\) is defined and \(\lim_{x \to a}g(x) = 0\), then \(\lim_{x \to a}f(x) = 0\) as well. Thus, if \(\lim_{x \to a}g(x) = 0\) but \(\lim_{x \to a}f(x) \neq 0\), then \(\lim_{x \to a}(f(x)/g(x))\) must be undefined.

Suppose that \(d > 0\), and for all \(x\), if \(0 < |x-a| < d\) then \(g(x) \leq f(x) \leq h(x)\). Suppose also that \(\lim_{x \to a} g(x) = \lim_{x \to a}h(x) = L\). Then \(\lim_{x \to a}f(x) = L\).

If \(\lim_{x \to a}|f(x)| = 0\), then \(\lim_{x \to a}f(x) = 0\).

There cannot be more than one number \(L\) such that as \(x \to a^{\neq}, f(x) \to L\).