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:**

**Negation:**

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