2
$\begingroup$

I want to show that the $\lim_{x\to 0}\frac{1}{x^2}$ does not exist.

I start with $|x-0|<\delta$ and $|\frac{1}{x^2}|<\epsilon$

I let $\epsilon = 1$ so $|\frac{1}{x^2}|<1$

Then $|1|<x^2$ which means $|1|<x$

If I make my $\delta= 1$

$|x-0|<1$

and $|x|<1$

I can't have $|x|<1$ and $|1|<x$ so I can't have a limit here. Is that right?

$\endgroup$
9
  • 1
    $\begingroup$ The limit exists and is $+\infty$. $\endgroup$ Dec 17, 2015 at 14:40
  • $\begingroup$ Thanks. I gathered that from my textbook. TO give a bit of context, the textbook (Stewart's calculus) is using this as a function that is not continous. It states that it is not continous because the limit does not exist at 0. So, I was wondering how I could show this. $\endgroup$
    – A nobody
    Dec 17, 2015 at 14:41
  • $\begingroup$ @EmilioNovati It is quite well possible that the OP is working in $\mathbb R$. In that context the limit does not exist. $\endgroup$
    – drhab
    Dec 17, 2015 at 14:42
  • $\begingroup$ I want to know why that is? Can it be shown in a straightforward way? $\endgroup$
    – A nobody
    Dec 17, 2015 at 14:42
  • 1
    $\begingroup$ Line number 2 assumes that $\delta$ is a very small positive number to indicate that $x \tends 0$. In that case, $\abs \frac{1}{x^2} < \epsilon$ has to be a very very large number and cannot be "let equal to 1" as you have done in your third line. That is where you have gone wrong. $\endgroup$ Dec 17, 2015 at 14:54

4 Answers 4

2
$\begingroup$

Suppose that the limit exists and equals $c\in\mathbb{R}$.

Then for e.g. $\epsilon>1$ some $\delta>0$ must exist with $\left|x\right|<\delta\implies\left|\frac{1}{x^{2}}-c\right|<1$.

However, if we take $\left|x\right|$ small enough then $\left|\frac{1}{x^{2}}-c\right|$ will definitely exceed $1$ (do you see why?).

We conclude that the limit does not exist.

$\endgroup$
2
  • $\begingroup$ Fantastic. Yes, I see why, as x goes below 1, the numerator starts to increase and will increase indefinitely as x decreases. $\endgroup$
    – A nobody
    Dec 17, 2015 at 15:01
  • $\begingroup$ Yes, you are correct. Glad to help. $\endgroup$
    – drhab
    Dec 17, 2015 at 15:04
2
$\begingroup$

Actually the limit exists and is equal to $\infty$. No matter from which direction you approach $0$ (from negative values or from positive values) the term $x^2$ is positive. Moreover as $x \to 0$, the term $x^2$ becomes very small, so that $\frac1{x^2}$ becomes very big (in other words, it grows steadily to infinity). Bringing all these together $$\lim_{x\to 0-}\frac1{x^2}=\lim_{x\to 0+}\frac{1}{x^2}=+\infty$$

$\endgroup$
3
  • $\begingroup$ Thanks Stef, I just realized that I did not pay attention to the "square" of the denominator and have promptly removed my mistaken answer. $\endgroup$ Dec 17, 2015 at 14:57
  • 3
    $\begingroup$ $\pm\infty$ are often not considered to be "real limits" as in "the limit is a real number". One might say that for $x\to 0$ the function is strictly divergent with $\lim\limits_{x\to 0} \frac{1}{x^2}=\infty$. $\endgroup$
    – Hirshy
    Dec 17, 2015 at 14:57
  • $\begingroup$ @Hirshy Yes, I agree with you. I just wanted to distinguish between "it doesn't exist" and "it goes to infinity". Thanks for correcting it. $\endgroup$
    – Jimmy R.
    Dec 17, 2015 at 14:58
2
$\begingroup$

By definition, the limit of a function at a point is defined as

$$\exists L:\left( {\forall \varepsilon > 0,\exists \delta > 0:\left( {\forall x,0 < \left| {x - a} \right| < \delta \to \left| {f(x) - L} \right| < \varepsilon } \right)} \right)$$

and its negation will be (see this post)

$$\forall L,\exists \varepsilon > 0:\left( \forall \delta > 0,\exists x:(0 < \left| x - a \right| < \delta \wedge |f(x) - L| \ge \varepsilon \right))$$

So for your special case we shall prove that

$$\forall L,\exists \varepsilon > 0: ( \forall \delta > 0,\exists x:(0 < \left| x \right| < \delta \wedge |\frac{1}{x^2} - L| \ge \varepsilon )$$

You can see this as a game! Your opponent chooses $L$ then you select an $\epsilon$. Next, the opponent chooses $\delta$ and then you choose the $x$. You should be wise enough to make proper choices so that you will win the game no matter what your opponent chooses. So your choices must satisfy the conditions $\left| x \right| < \delta \wedge |\frac{1}{x^2} - L| \ge \varepsilon$ regardless of opponent's choices. So, let us play!

The opponent makes a choice for $L$. we choose $\varepsilon$ to be $1$. Then the opponent selects a $\delta$ and we should make our final choice for $x$ such that

$$\left| x \right| < \delta \wedge |\frac{1}{x^2} - L| \ge 1$$

we can choose $x_n=\frac{1}{\sqrt{n+L}}$ and our conditions turns to be

$$\left| \frac{1}{\sqrt{n+L}} \right| < \delta \wedge |n| \ge 1$$

So, no matter what the opponent chooses, we will select large enough values for $n$ such that the above conditions are satisfied and we win the game! Hence, the limit does not exist!

$\endgroup$
1
$\begingroup$

Find the following limit: $$\lim _{x \rightarrow 0} \frac{1}{x^{2}}$$ $$\lim _{x \rightarrow 0} \frac{1}{x^{2}}=\lim _{x \rightarrow 0} \exp \left(\log \left(\frac{1}{x^{2}}\right)\right):$$ $$\lim _{x \rightarrow 0} \exp \left(\log \left(\frac{1}{x^{2}}\right)\right)$$ $$\exp \left(\log \left(\frac{1}{x^{2}}\right)\right)=\exp (-2 \log (x)):$$ $$\lim _{x \rightarrow 0} \exp (-2 \log (x))$$ $$\lim _{x \rightarrow 0} \exp (-2 \log (x))=\exp \left(\lim _{x \rightarrow 0}-2 \log (x)\right):$$ $$\exp \left(\lim _{x \rightarrow 0}-2 \log (x)\right)$$ Applying the product rule, write $\lim _{x \rightarrow 0}-2 \log (x)$ as $-2 \lim _{x \rightarrow 0} \log (x)$ : $$\exp \left(-2 \lim _{x \rightarrow 0} \log (x)\right)$$ $$\lim _{x \rightarrow 0} \log (x)=-\infty:$$ $$e^{-2[-\infty}$$ $$e^{-2(-\infty)}=\infty:$$ $$\lim_{x\to 0}\frac{1}{x^2}: \infty$$

$\endgroup$

You must log in to answer this question.

Not the answer you're looking for? Browse other questions tagged .