Limits

Notation

${\lim }_{x\to a}f(x)=L$ The value of f(x) can be made arbitrarily close to L by choosing x sufficiently close to a but not equal to a

Definitions

• Limits Formalized Definition
• Limits Open Set Definition

Limit Properties

You require the properties of limits to be able to get the limits of any function https://en.wikibooks.org/wiki/Calculus/Proofs_of_Some_Basic_Limit_Rules

Limits are Unique

Limits Determined by Component Functions

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 ${\lim }_{x\to a}[f(x{)}^n]={\lim }_{x\to a}[f(x){]}^n$

Composition Rule

If $f$ is continuous then ${\lim }_{x\to a}f(g(x))=f({\lim }_{x\to a}g(x))$

Concepts

• Discontinuities
• Continuous
• Limits Does Not Exist
• Order Limit Law
• Limit Existence