Skip to content

Limit

Limit Laws

If \(L, M, c, k \in \mathbb{R}\) and \(\lim\limits_{x \to c} f(x) = L\) and \(\lim\limits_{x \to c} g(x) = M\), then

  1. Sum Rule: \(\lim\limits_{x \to c} (f(x) + g(x)) = L + M\)
  2. Difference Rule: \(\lim\limits_{x \to c} (f(x) - g(x)) = L - M\)
  3. Constant Multiple Rule: \(\lim\limits_{x \to c} (k \cdot f(x)) = k \cdot L\)
  4. Product Rule: \(\lim\limits_{x \to c} (f(x) \cdot g(x)) = L \cdot M\)
  5. Quotient Rule: \(\lim\limits_{x \to c} (\frac{f(x)}{g(x)}) = \frac{L}{M}, M \not= 0\)
  6. Power Rule: \(\lim\limits_{x \to c} [f(x)]^n = L^n, n \in N^*\)
  7. Root Rule: \(\lim\limits_{x \to c} \sqrt[n]{f(x)} = \sqrt[n]{L}\)

Theorems

\(\lim\limits_{x \to 0} \dfrac{\sin x}{x} = \lim\limits_{x \to 0} \dfrac{x}{x} = 1\)

Rational Function

  1. \(\lim\limits_{x \to \infty} \frac{higher\ degree}{lower\ degree} = 0\)
  2. \(\lim\limits_{x \to \infty} \frac{lower\ degree}{higher\ degree} = DNE\)
  3. \(\lim\limits_{x \to 0} \frac{same\ degree}{same\ degree} = \frac{highest\ degree\ coefficient}{highest\ degree\ coefficient}\)