To say that as \(x \to a^{<}, f(x) \to L\) means that for every \(\epsilon > 0\) there is some \(\delta > 0\) such that if \(a-\delta < x < a\) then \(|f(x)-L| < \epsilon\). To say that as \(x \to a^{>}, f(x) \to L\) means that for every \(\epsilon > 0\) there is some \(\delta > 0\) such that if \(a < x < a+\delta\) then \(|f(x)-L| < \epsilon\).

If there is a number \(L\) such that as \(x \to a^{-}, f(x) \to L\), then we defined \(\lim_{x \to a^{-}} f(x)\) to be the unique such number \(L\). If there is no such number \(L\), then \(\lim_{x \to a^-} f(x)\) is undefined. Similarly, \(\lim_{x \to a^+} f(x)\) is the unique number \(L\) such that as \(x \to a^+, f(x) \to L\), if there is such a number \(L\).

Suppose that \(d > 0\), and \(f_1\) and \(f_2\) are functions such that for all \(x\), if \(a-d < x < a\) then \(f_1(x) = f_2(x)\). Then \(\lim_{x \to a^-} f_1(x) = \lim_{x \to a^-}f_2(x)\).

For any function \(f\) and numbers \(a\) and \(L\), \(\lim_{x \to a} f(x)=L\) if and only if both \(\lim_{x \to a^-}f(x)=L\) and \(\lim_{x \to a^+}f(x) = L\).

To say that as \(x \to \infty\), \(f(x) \to L\) means that for every \(\epsilon > 0\) there is some N such that if \(x > N\) then \(|f(x) - L| < \epsilon\). We defined \(\lim_{x \to \infty} f(x)\) to be the unique number \(L\) such that as \(x \to \infty\), \(f(x) \to L\), if there is such a number L.

(Here we can think \(\epsilon\) as challenge and \(N\) as the reponse for it)

To say that as \(x \to -\infty\), \(f(x) \to L\) means that for every \(\epsilon > 0\) there is some N such that if \(x < N\) then \(|f(x) - L| < \epsilon\). We defined \(\lim_{x \to -\infty} f(x)\) to be the unique number \(L\) such that as \(x \to -\infty\), \(f(x) \to L\), if there is such a number L.

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

1. If \(\lim_{x \to a}f(x) = \infty\) and either \(\lim_{x \to a}g(x) = L\) or \(\lim_{x \to a}g(x) = \infty\), then \(\lim_{x \to a} f(x) + g(x) = \infty\)
2. If \(\lim_{x \to a}f(x) = -\infty\) and either \(\lim_{x \to a}g(x) = L\) or \(\lim_{x \to a}g(x) = -\infty\), then \(\lim_{x \to a} f(x) + g(x) = -\infty\)
3. If \(\lim_{x \to a}f(x) = \infty\) and either \(\lim_{x \to a}g(x) = L > 0\) or \(\lim_{x \to a}g(x) = \infty\), then \(\lim_{x \to a} f(x) . g(x) = \infty\)
4. If \(\lim_{x \to a}f(x) = \infty\) and either \(\lim_{x \to a}g(x) = L < 0\) or \(\lim_{x \to a}g(x) = -\infty\), then \(\lim_{x \to a} f(x) . g(x) = -\infty\)
5. If \(\lim_{x \to a}f(x) = -\infty\) and either \(\lim_{x \to a}g(x) = L > 0\) or \(\lim_{x \to a}g(x) = \infty\), then \(\lim_{x \to a} f(x) . g(x) = -\infty\)
6. If \(\lim_{x \to a}f(x) = -\infty\) and either \(\lim_{x \to a}g(x) = L < 0\) or \(\lim_{x \to a}g(x) = -\infty\), then \(\lim_{x \to a} f(x) . g(x) = \infty\)