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Calculus - Limit Laws

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Sum Law for Limits

\lim_{{x \to c}} f(x) = L

and

\lim_{{x \to c}} g(x) = M

, then

\lim_{{x \to c}} [f(x) + g(x)] = L + M

Product Law for Limits

\lim_{{x \to c}} f(x) = L

and

\lim_{{x \to c}} g(x) = M

, then

\lim_{{x \to c}} [f(x) \cdot g(x)] = L \cdot M

Quotient Law for Limits

\lim_{{x \to c}} f(x) = L

and

\lim_{{x \to c}} g(x) = M

where

M \neq 0

, then

\lim_{{x \to c}} \frac{{f(x)}}{{g(x)}} = \frac{L}{M}

Power Law for Limits

\lim_{{x \to c}} f(x) = L

and

n

is a positive integer, then

\lim_{{x \to c}} [f(x)]^n = L^n

Root Law for Limits

\lim_{{x \to c}} f(x) = L

and

n

is a positive integer such that the expression is well-defined, then

\lim_{{x \to c}} \sqrt[n]{{f(x)}} = \sqrt[n]{L}

Difference Law for Limits

\lim_{{x \to c}} f(x) = L

and

\lim_{{x \to c}} g(x) = M

, then

\lim_{{x \to c}} [f(x) - g(x)] = L - M

Constant Multiple Law for Limits

\lim_{{x \to c}} f(x) = L

and

k

is a constant, then

\lim_{{x \to c}} [k \cdot f(x)] = k \cdot L

Continuous Function Law for Limits

If $f(x)$ is continuous at $x = c$ , then

\lim_{{x \to c}} f(x) = f(c)

Compound Function Law for Limits

\lim_{{x \to c}} g(x) = L

and

f(x)

is continuous at

L

, then

\lim_{{x \to c}} f(g(x)) = f(L)

Squeeze Theorem

f(x) \leq g(x) \leq h(x)

for all

x

near

c

, and

\lim_{{x \to c}} f(x) = \lim_{{x \to c}} h(x) = L

, then

\lim_{{x \to c}} g(x) = L

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