We know that the statement

$$\lim\limits_{x\to \infty}f(x)$$ exsixts has the following meaning.Negate the statement. $$\exists(\forall A)(\forall t > 0) (\exists \delta > 0 ) (\forall x) ( 0 < |x – a| < \delta \Rightarrow |f(x) – A| < \varepsilon$$

Solution:

The negative of the statement is

$\forall(\exists A) ( \exists t > 0) (\forall \delta > 0) (\exists x) (0 < |x – a| < \delta \Rightarrow |f(x) – A| < \varepsilon$

Write the following statement in symbol form and then form its negation, for each $$\varepsilon > 0$$, there exsists an N > 0 s.t $$| fn(x) – f(x) | < ϵ \ Whenever\ n > N and\ S x \in S.$$

Solution:

$\forall \varepsilon > 0,\ \exists N > 0$ $s.t\ |fn(x) – f(x) | < \varepsilon$

Negation:

$\exists \varepsilon > 0, \ \forall N > 0 \\ s.t\ |fn(x) – f(x) | > \varepsilon\\ \ whenever\ n > N \ and\ x \in S.$

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