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$f$ is a function with the domain of all real numbers, and let $a$, $L$ be elements of $\Bbb R$.

If $\lim_{x \rightarrow a}f(x) = L$, then $\lim_{x \rightarrow a/5}f(5x)=L$

How would you do this without the limit law theorems?

Ivo Terek
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James
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1 Answers1

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Take $\epsilon > 0$. We want $\delta > 0$ such that $$0< |x - a/5| <\delta \implies |f(5x)-L| < \epsilon,$$right? How do we find the little bastard? He's hidden inside the limit that we already know that exists. For that fixed $\epsilon$, there is $\delta' > 0$ satisfying $$0 < |t-a|<\delta'\implies |f(t) - L|<\epsilon.$$I used $t$ instead of $x$ on purpose. With this, put $\delta \doteq \delta'/5>0$. Then: $$0< |x-a/5| < \delta \implies 0 < |5x-a|<5\delta=\delta' \implies |f(5x)-L|<\epsilon,$$as wanted. The moral of the history is that $5x$ plays the role of $t$ on the limit we already know that holds. My other answer here might also be useful.

Ivo Terek
  • 77,665
  • What does $δ≐δ′/5>0$ mean? I'm not really familiar with the symbol $≐$. – James Oct 05 '17 at 01:57
  • The dot means I am defining $\delta$ as $\delta'/5$. Perhaps you've seen the equivalent $\delta := \delta'/5$ before? – Ivo Terek Oct 05 '17 at 01:58
  • Make $t = 5x$ and look at the first implication in display, which is true by hypothesis. – Ivo Terek Oct 05 '17 at 02:17
  • Yea my mistake, should i re-enter the question for clarity for future readers? – James Oct 05 '17 at 02:17
  • No, I don't see any problem. The question seems clear as it can get. – Ivo Terek Oct 05 '17 at 02:18
  • One more thing, was changing the variable necessary or was that just to make the answer clearer? Like, could they have both been x or would they have to be x and x'? – James Oct 05 '17 at 02:21
  • Just to make it clearer for you. I could have used $x$ everywhere, since it would be a "dummy" variable. It boils down to logic. – Ivo Terek Oct 05 '17 at 02:23
  • I was looking over this question again and I was wondering why you were able to set x ('t') to 5x, I thought x was fixed to be all real numbers by definition? – James Oct 07 '17 at 02:50
  • Again: $x$ is a dummy variable, just as in $$\sum_i a_i = \sum_j a_j$$ or $$\int_0^1f(u);{\rm d}u = \int_0^1f(v);{\rm d}v$$or $$\lim_{x\to a}f(x) = \lim_{t\to a} f(t). $$ – Ivo Terek Oct 07 '17 at 02:54
  • But t = 5x which =/= x right? – James Oct 07 '17 at 02:59