# Limits

Limits Fact Sheet

# Formal Definition of Limit

{$$ \begin{gather} \lim_{x \rightarrow a} f(x) = L \iff \cr \forall \epsilon \in \mathbb R > 0, \exists \delta \in \mathbb R, \text{ such that} \cr 0 < |x-a| < \delta \implies |f(x) - L| < \epsilon \end{gather} $$}

Notice it is not necessary that {$f(a)$} exists, and {$f(a)$} appears nowhere in the definition.

The usual method of proof of a limit is to find an expression for δ in terms of ε thus ensuring a δ exists for every ε.

# Right and Left Limits

## Right hand limit

If x approaches a from the right (from above):

{$$ \lim_{x \rightarrow a^+} f(x) = R $$}

## Left hand limit

If x approaches a from the left (from below):

{$$ \lim_{x \rightarrow a^-} f(x) = L $$}

*Sources:*

- Kouba, Duane.
*Precise Definition of Limit*(University of California at Davis).(approve sites) - Bernard Bolanzo Wikipedia
- Karl Weierstrass Wikipedia

** Recommended:** Bernard Bolanzo Topic Karl Weierstrass Topic

** Categories:** Calculus Topice

** Tags:** XxxLimit XxxPreCalculus XxxAlgebra