Limit Laws and Properties

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Limit of a Constant:

$\Large \lim_{x \to a} c = c$

Limit of Single Variable:

$\Large \lim_{x \to a} x = a$

If the Function is Continuous:

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

The Constant Multiple Law:

$\Large \lim_{x \to a} [c f(x)] = c \lim_{x \to a} f(x)$

The Sum and Difference Law:

$\Large \lim_{x \to a} [f(x) \pm g(x)] = \lim_{x \to a} f(x) \pm \lim_{x \to a} g(x)$

The Product Law:

$\Large \lim_{x \to a} [f(x)g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$

The Quotient Law:

$\Large \lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}, \quad \lim_{x \to a} g(x) \ne 0$

The Power Law:

$\Large \lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n, \quad n \in \mathbb{N}$

The Root Law:

$\Large \lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}, \quad n \in \mathbb{N}$

Exponential Law:

$\Large \lim_{x \to a} a^{f(x)} = a^{\lim_{x \to a} f(x)}$

Case 1 (n > m):

$\Large \lim_{x \to \infty} \frac{x^m + x^{m-1} + \cdots}{x^n + x^{n-1} + \cdots} = 0$

Case 2 (n < m):

$\Large \lim_{x \to \infty} \frac{x^m + x^{m-1} + \cdots}{x^n + x^{n-1} + \cdots} = \infty$

Case 3 (n = m):

$\Large \lim_{x \to \infty} \frac{a x^m + \cdots}{b x^n + \cdots} = \frac{a}{b}$

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