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. $\lim\limits_{x \to 0} \frac{same\ degree}{same\ degree} = \frac{highest\ degree\ coefficient}{highest\ degree\ coefficient}$

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