Please advise: is my proof sufficient? or do I need to use $\epsilon$?

Prove that $$ \lim_{x \to \infty} \ f(x)= \lim_{t \to 0^+} \ f(1/t)$$ if these limits exist

Proof: putting $t=\dfrac{1}{x}$ then as $x\to\infty, \ t\to0^+$ and $x=\dfrac{1}{t}$

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

  • 5
    $\begingroup$ While what you wrote is indeed the idea behind what's happening, I would be willing to bet money that whoever gave you that assignment expects you to use $\epsilon$. $\endgroup$
    – Arthur
    Sep 7, 2017 at 13:12
  • 1
    $\begingroup$ Does this answer your question? Limit of $f(x)$ and $f(1/x)$? $\endgroup$
    – user53259
    Jan 2, 2022 at 7:40

1 Answer 1


This is just exercise 75 in Section 2.7, James Stewart, Calculus Early Transcendentals 7th ed. 2011. Page 143.

Note that this result is true for $x \rightarrow -\infty$ and $t \rightarrow 0^-$ also.

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