Complete calculus formula sheet

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Limits & Continuity

Limits & Continuity
Formula	Statement	Conditions
Limit Definition (Epsilon-Delta)	$\lim_{x \to a} f(x) = L \iff \forall \epsilon > 0, \exists \delta > 0 : 0 < \|x - a\| < \delta \Rightarrow \|f(x) - L\| < \epsilon$	f(x) must be defined on an open interval containing a, except possibly at a itself.
Left-Hand Limit	$\lim_{x \to a^-} f(x) = L$	f(x) must be defined on an open interval (c, a) for some c < a.
Right-Hand Limit	$\lim_{x \to a^+} f(x) = L$	f(x) must be defined on an open interval (a, d) for some d > a.
Squeeze Theorem	$g(x) \leq f(x) \leq h(x) \text{ and } \lim_{x \to a} g(x) = \lim_{x \to a} h(x) = L \Rightarrow \lim_{x \to a} f(x) = L$	no restriction stated
Limit of a Sum	$\lim_{x \to a} [f(x) + g(x)] = \lim_{x \to a} f(x) + \lim_{x \to a} g(x)$	Both lim f(x) and lim g(x) must exist.
Limit of a Product	$\lim_{x \to a} [f(x) \cdot g(x)] = \lim_{x \to a} f(x) \cdot \lim_{x \to a} g(x)$	Both lim f(x) and lim g(x) must exist.

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