real-analysis Limit property question $\text{Here is a question that I have been given, }$
$\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,  &amp; = |\theta-(a-c)| \lt\ \delta \\
&amp; =|(x-c)-(a-c)| \lt\ \delta \\
&amp; =|x-a| \lt\ \delta\\
&amp; \implies\ |f(x)-A|\lt\epsilon\\&amp;\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 $c\neq0,\lim_{x\to a/c}f(cx)=A$.
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.
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.