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I am confused regarding one sided limits and how to calculate it.

For Example: $$\lim_{x\to 0}\frac{1}{x}\quad\text{does not exist}$$ How can I validate that $\lim\limits_{x\to 0^+}\frac{1}{x}$ or $\lim\limits_{x\to 0^-}\frac{1}{x}$ exists?

I am pretty certain that the limits do exist because if we take a positive value which is small $0.0000000\dots1$ we will get a very big positive limit value. And the same for the negative value. I know that in this case RHL$\ne$LHL.

I hope somebody can help me figure this one out. Thank you.

Sorry could not write down the equations, hope my explanation is clear enough.

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  • $\begingroup$ You are right. The limits equal positive and negative infinity. $\endgroup$
    – user142299
    Apr 21, 2014 at 4:58
  • $\begingroup$ I wanted to use some mathematical derivation to prove the same. Hope you can help. $\endgroup$ Apr 21, 2014 at 5:01

3 Answers 3

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$\mathbf{Definition} $:

$$ \boxed{ \lim_{x \to a^+ } f(x) = \infty } $$

means that for all $\alpha > 0$, there exists $\delta > 0$ such that if $ 0<x -a < \delta$, then $f(x) > \alpha$

$\mathbf{Example} $:

$$ \lim_{x \to 0^+} \frac{1}{x} = \infty $$

We use the definition to establish this fact. In other words, for a given $\alpha > 0$, we need to find a $\delta > 0$ such that if $0< x < \delta $, then $\frac{1}{x} > \alpha $

Notice $$ \frac{1}{x} > \alpha \iff \frac{1}{\alpha} > x$$

Hence, if we select $\delta = \frac{1}{\alpha } $, we will achieve our desired result.

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To say that $\frac1x \to \infty$ as $x \to 0^+$ (the plus means goes to $0$ from the positive direction) is to say that for any large $N > 0$ I can choose a small $\epsilon > 0$ (whose choice depends on $N$) such that $0 < x < \epsilon$ implies $\frac1x > N$. This, of course, is achieved by choosing $\epsilon = \frac1N$.

I'll leave the negative side to you, it's almost identical.

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From a bit of intuition, you can see:

$$\lim_{x \to 0^+}(1/x) = \infty$$ $$\lim_{x \to 0^-}(1/x) = -\infty$$

Of course, this is because as the value for $x$ gets infinitesimally small, the fraction blows up. To actually prove these limits in a rigorous fashion, however, you'd need to do the following:

For the first limit: Given any $n \in \mathbb{R}$, show that there exists an $\epsilon > 0$ such that, for all $0<x<\epsilon$, we'll have $1/x > n$.

And you'd proceed likewise for the second.

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  • $\begingroup$ $n$ should be positive $\endgroup$
    – user139708
    Apr 21, 2014 at 5:11
  • $\begingroup$ I don't think it matters. $\mathbb{R}$ includes all positive real numbers as well. Of course, it is also sufficient to restrict $n$ to the positives. $\endgroup$
    – Kaj Hansen
    Apr 21, 2014 at 5:13

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