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I read the theory of limits and i have some misunderstanding. For example we have simple limit expression:

$$\lim _{x\rightarrow \infty}{\frac{1}{x}}$$

I see that this limit is 0 and if build graph of this sequence we see that limit is 0. But i only see it, is's only my guesses... How we can find solution for this analytical? How can we solve it?

Thank you.

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up vote 2 down vote accepted

The definition of the limit as $x \to \infty$ of a function is:

$\lim_{x \to \infty} f(x)=L \in \Bbb{R}$ if $\forall \varepsilon >0$ there exists $\delta>0$ such that for every $x > \delta$ we have $f(x) \in (L-\varepsilon,L+\varepsilon)$

Pick $\varepsilon>0$. Then there exists a real number $S> \frac{1}{\varepsilon}$, i.e. $\frac{1}{S} <\varepsilon$. Pick $\delta =S$ and you are done.

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