Limit Properties

Because limits have the unique ability to determine values at discontinuities, it is much more efficient to evaluate with limits than regular substitution. https://en.wikibooks.org/wiki/Calculus/Proofs_of_Some_Basic_Limit_Rules Before using limit properties, you must show that both limits ${\lim }_{x\to c}f(x)=L$ and ${\lim }_{x\to c}g(x)=M$ exist

Identity Rule

${\lim }_{x\to a}x=a$

Constant Rule

if $a$ and $k$ for any constants then ${\lim }_{x\to a}k=k$

Sums & Difference Rule

The limit of the sum, is the sum of the limits. ${\lim }_{x\to a}[f(x)\pm g(x)]={\lim }_{x\to a}[f(x)]\pm {\lim }_{x\to a}[g(x)]$

Scalar Product Rule

${\lim }_{x\to a}[cf(x)]=c[{\lim }_{x\to a}f(x)]$

Product Rule

${\lim }_{x\to a}[f(x)g(x)]={\lim }_{x\to a}[f(x)]\ast {\lim }_{x\to a}[g(x)]$

Quotient Rule

${\lim }_{x\to a}\left[\frac{f(x)}{g(x)}\right]=\frac{{\lim }_{x\to a}[f(x)]}{{\lim }_{x\to a}[g(x)]}$

Power Rule

Limit of the power is the power of the limit ${\lim }_{x\to a}[f(x{)}^{n]}=[{\lim }_{x\to a}f(x){]}^n$