we have been doing epsilon and delta limit proofs in my first year calculus class and I can't seem to wrap my around this stuff. We have not done an example in class of disproving a limit, and this was given as a practice question for my exam so I really need some help, Thank You!

Prove that the $\lim_{x \rightarrow 0}\frac1x$ does not exist

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    $\begingroup$ $$\lim_{x\to0^+}\frac1x=+\infty, \lim_{x\to0^-}\frac1x=-\infty$$ $\endgroup$ Dec 10, 2013 at 10:56
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    $\begingroup$ Yeah I unnderstand that, but I need to prove it using the definition of a limit (epsilons and deltas) Thanks! $\endgroup$ Dec 10, 2013 at 10:57

1 Answer 1


Show that no number $\alpha \in \mathbb R$ is a limit of $\frac 1x$ for $x \to 0$. That is show that $$ \exists \epsilon > 0 \; \forall \delta > 0 \; \exists x:\; x \in (-\delta, \delta) \land\left|\frac 1x - \alpha\right| > \epsilon $$ Let $\epsilon = 1$ and given $\delta$, set $x = \min\{\frac\delta 2, \frac 1{|\alpha| + 2}\}$.


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