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What are the rules for evaluating limits at infinity.

I don't know if l can apply the limits law for $\infty-\infty$ or $\infty/\infty$ etc.

The value of $\lim x - \lim x$ as $x \to \infty$ is undefined ( infinity).

The value of $\lim (x-x)= \lim 0$ as $x\to \infty$ is zero.

Can we apply the usual laws of limit here ?

user
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Aristotle Stagiritis
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2 Answers2

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The addition theorem states that

$$\lim_{x\to a}(f(x)\pm g(x))=\lim_{x\to a}f(x)\pm \lim_{x\to a}g(x)$$

if these limits exist.

Hence

$$\lim_{x\to\infty}(x-x)\not\equiv\lim_{x\to\infty}x-\lim_{x\to\infty}x$$

raises no contradiction. (In fact this pseudo-paradox is not related to the variable going to infinity, it is due to the function going to infinity.)

Limits to infinity are not essentially different from finite limits. You can always transform

$$\lim_{x\to\infty}f(x)=\lim_{t\to 0}f\left(\frac1{|t|}\right).$$

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    @user: I could tell you, but the reference is too large to fit in this margin. –  Nov 26 '19 at 09:34
  • This may not be directly related to the question but regarding infinity-infinity, can it converge to zero ? Consider A=B to be the sum of natural numbers. Is A-B=0 ? – Aristotle Stagiritis Nov 26 '19 at 09:42
  • @AristotleStagiritis: this is answered in my post. –  Nov 26 '19 at 10:19
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Given

  • $f(x) \to \infty$

  • $g(x) \to \infty$

  • $h(x) \to L>0$

we can always conclude that

$$f(x)+g(x)\to \infty, \quad f(x)\cdot (\pm g(x))\to \pm\infty$$

$$f(x)\pm h(x)\to \infty, \quad f(x)\cdot (\pm h(x))\to \pm\infty$$

and more in general we can apply similar rules when the expression is not in an indeterminate form.

With reference to your example, you are right

$$\lim_{x\to \infty} x -\lim_{x\to \infty} x$$

is undefined but

$$\lim_{x\to \infty} (x-x)=\lim_{x\to \infty} 0=0$$

Refer also to the related

user
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