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$\text{Here is a question that I have been given, }$ $\lim_{x\to a}f(x)=A \iff $for each $c\in\mathbb R,\lim_{x\to a-c}f(x+c)=A$.

For this question the proof follows:

$\text{ Proof: Assume } \lim_{\theta\to a}f(x)=A; \ \\\forall\epsilon\gt0,\exists\delta\gt0,\forall{\theta}\in\mathbb R, 0\lt | \theta-a| \lt\delta\implies\ |f(x)-A|\lt\epsilon.\\ \text{Given }\epsilon\gt0, \\\text{If$\;$ } 0\lt\; |\theta-(a-c)| \lt\ \delta, \text{choose$\;$ } \theta=x-c,$ $ $\begin{equation} \begin{split} \text{Then$\;$} 0\lt\; |\theta-(a-c)| \lt\ \delta, & = |\theta-(a-c)| \lt\ \delta \\ & =|(x-c)-(a-c)| \lt\ \delta \\ & =|x-a| \lt\ \delta\\ & \implies\ |f(x)-A|\lt\epsilon\\&\implies\lim_{\theta\to a-c}f(x+c)=A \end{split} \end{equation}

$\text{$\therefore$ }$ $\lim_{x\to a}f(x)=A \iff $for each $c\in\mathbb R,\lim_{x\to a-c}f(x+c)=A$.

Prove or disprove: Now this one I have tried similar method's with no success.I can't figure what $\theta$ should be in this example.

$\lim_{x\to a}f(x)=A \iff $for each $c\neq0,\lim_{x\to a/c}f(cx)=A$.

math4fun
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  • Hi, please show us what you have done. – meta_warrior Nov 17 '13 at 08:54
  • I have tried many variation changing the variable theta around with theta=a/c, I have tried c*theta=a and tried to solve, c=a/theta etc. It took me forever to input what I have since I am new to LaTex. I will work on entering in my scratch work on the second one so far – math4fun Nov 17 '13 at 10:23

1 Answers1

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You can easily solve your problem by applying the theorem stated by Pedro Tamaroff in this thread. Roughly speaking, you change variable: $cx=u$, and remark that $u \to a$ whenever $x \to a/c$. All this seems rather obvious, but it isn't.

Community
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Siminore
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  • I have tried many variation changing the variable theta around with theta=a/c, I have tried c*theta=a and tried to solve, c=a/theta etc. It took me forever to input what I have since I am new to LaTex. I will work on entering in my scratch work on the second one so far – math4fun Nov 17 '13 at 10:29