# The Negation

• October 22nd 2009, 12:03 PM
ksbartlett
The Negation
The Question:
Rewrite the definition in terms for logical symbols, then write the negation using the same symbols.

The real number L is the limit of the function $f:D \rightarrow \mathbb{R}$ at the point c iff for each $\varepsilon > 0$ there exists $\delta > 0$ such that $|f(x) - L|< \varepsilon$ whenever $x \in D$ and $0 < |x - c| < \delta$.

$\forall \varepsilon > 0, \exists \delta > 0, ( x \in D \cap 0 < |x - c| < \delta )\Rightarrow |f(x) - L| < \varepsilon$
$\exists \varepsilon > 0, \forall \delta > 0, ( x \in D \cap 0 < |x - c| < \delta ) \cap |f(x) - L| \geq \varepsilon$
$\forall\,\epsilon>0,\,\exists\,\delta>0,\,\forall\ ,x\in D,\,0<|x-c|<\delta\ \Rightarrow\ |f(x)-L|<\epsilon$