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# Limits

Bernard Bolanzo

Karl Weierstrass

#### Contents

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:

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

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