Limits Rules

Basic Limit Laws

Power and Root Limits

Special Theorems

Trigonometric Limits

Exponential and Logarithmic Limits

Basic Limit Laws

	law	formula	explanation
	Constant Rule	$\lim_{x \to a} c = c$	The limit of a constant is the constant itself
	Identity Rule	$\lim_{x \to a} x = a$	The limit of x as x approaches a is a
	Sum Rule	$\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$	The limit of a sum is the sum of the limits
	Difference Rule	$\lim_{x \to a} [f(x) - g(x)] = \lim_{x \to a} f(x) - \lim_{x \to a} g(x)$	The limit of a difference is the difference of the limits
	Constant Multiple Rule	$\lim_{x \to a} [c \cdot f(x)] = c \cdot \lim_{x \to a} f(x)$	Constants can be factored out of limits
	Product Rule	$\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$	The limit of a product is the product of the limits
	Quotient Rule	$\lim_{x \to a} \frac{f(x)}{g(x)} = \frac{\lim_{x \to a} f(x)}{\lim_{x \to a} g(x)}$	The limit of a quotient is the quotient of limits, provided denominator limit is not zero

Power and Root Limits

	law	formula	explanation
	Power Rule	$\lim_{x \to a} [f(x)]^n = [\lim_{x \to a} f(x)]^n$	The limit of a power is the power of the limit
	Root Rule	$\lim_{x \to a} \sqrt[n]{f(x)} = \sqrt[n]{\lim_{x \to a} f(x)}$	The limit of a root is the root of the limit, provided the root exists
	Polynomial Rule	$\lim_{x \to a} P(x) = P(a)$	For polynomial P(x), substitute the value directly
	Rational Function Rule	$\lim_{x \to a} \frac{P(x)}{Q(x)} = \frac{P(a)}{Q(a)}$	For rational functions, substitute directly if denominator is not zero

Special Theorems

	law	formula	explanation
	Squeeze Theorem	If $g(x) \leq f(x) \leq h(x)$ and $\lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L$ , then $\lim_{x \to a} f(x) = L$	Function squeezed between two functions with same limit has that limit
	Absolute Value Rule	$\lim_{x \to a} \|f(x)\| = \|\lim_{x \to a} f(x)\|$	The limit of absolute value is absolute value of limit, if limit exists
	Composition Rule	If $\lim_{x \to a} g(x) = L$ and $f$ is continuous at $L$ , then $\lim_{x \to a} f(g(x)) = f(L)$	Limit of composition when inner function has limit and outer is continuous

Trigonometric Limits

	law	formula	explanation
	Fundamental Sine Limit	$\lim_{x \to 0} \frac{\sin x}{x} = 1$	The fundamental trigonometric limit for sine
	Fundamental Cosine Limit	$\lim_{x \to 0} \frac{1 - \cos x}{x} = 0$	The fundamental trigonometric limit for cosine
	Alternative Cosine Limit	$\lim_{x \to 0} \frac{1 - \cos x}{x^2} = \frac{1}{2}$	Alternative form of the cosine limit
	Tangent Limit	$\lim_{x \to 0} \frac{\tan x}{x} = 1$	The fundamental trigonometric limit for tangent

Exponential and Logarithmic Limits

	law	formula	explanation
	Natural Exponential Definition	$\lim_{x \to \infty} \left(1 + \frac{1}{x}\right)^x = e$	The definition of e as a limit
	Alternative e Definition	$\lim_{x \to 0} (1 + x)^{\frac{1}{x}} = e$	Alternative form for the definition of e
	Exponential Limit	$\lim_{x \to 0} \frac{e^x - 1}{x} = 1$	Fundamental limit involving natural exponential
	General Exponential Limit	$\lim_{x \to 0} \frac{a^x - 1}{x} = \ln a$	Fundamental limit for exponential with base a
	Natural Logarithm Limit	$\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1$	Fundamental limit involving natural logarithm
	General Logarithm Limit	$\lim_{x \to 0} \frac{\log_a(1 + x)}{x} = \frac{1}{\ln a}$	Fundamental limit for logarithm with base a