Bernard Bolanzo
Bernard Bolanzo
Karl Weierstrass
Karl Weierstrass


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):

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


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

Recommended: Bernard Bolanzo Topic Karl Weierstrass Topic

Categories: Calculus Topice

Tags: XxxLimit XxxPreCalculus XxxAlgebra


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