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Limit

Limit Laws

If L,M,c,k∈RL, M, c, k \in \mathbb{R} and lim⁡x→cf(x)=L\lim\limits_{x \to c} f(x) = L and lim⁡x→cg(x)=M\lim\limits_{x \to c} g(x) = M, then

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

Theorems

lim⁡x→0sin⁡xx=lim⁡x→0xx=1\lim\limits_{x \to 0} \dfrac{\sin x}{x} = \lim\limits_{x \to 0} \dfrac{x}{x} = 1

Rational Function

  1. lim⁡x→∞higher degreelower degree=0\lim\limits_{x \to \infty} \frac{higher\ degree}{lower\ degree} = 0
  2. lim⁡x→∞lower degreehigher degree=DNE\lim\limits_{x \to \infty} \frac{lower\ degree}{higher\ degree} = DNE
  3. lim⁡x→0same degreesame degree=highest degree coefficienthighest degree coefficient\lim\limits_{x \to 0} \frac{same\ degree}{same\ degree} = \frac{highest\ degree\ coefficient}{highest\ degree\ coefficient}