Proving: $\lim _{x \rightarrow \infty} \ f(x)=\lim_{t \rightarrow 0^+} \ f(1/t)$

Question

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

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 $\epsilon$. — Arthur, Sep 07 '17 at 13:12
Does this answer your question? Limit of $f(x)$ and $f(1/x)$? — , Jan 02 '22 at 07:40

score 0 · Answer 1 · answered Jan 12 '21 at 21:45

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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.

answered Jan 12 '21 at 21:45

1 Answers1