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Given constants $a>0$, $b>0$, and a function $f(x)$ for which $\lim_{x \rightarrow \infty} \ f(x) = a$. Suppose we have another function $g(x)$ for which $g(bx) = f(x)$. Can we compute $\lim_{x \rightarrow \infty} \ g(x)$ based on the given information? This question is distilled from an integral problem.

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any info on the sign of $b$? – Maesumi Oct 8 '12 at 16:59
up vote 0 down vote accepted

We know that $g(x)=f(x/b)$.

$\newcommand{\limti}[1]{\lim\limits_{#1\to\infty}}\limti x f(x)=a$ means that for each $\varepsilon>0$ there exists $x_0$ such that $$x\ge x_0 \Rightarrow |f(x)-a|<\varepsilon.$$ Thus for $x\ge bx_0$ we have $x/b \ge x_0$ and $|g(x)-a|=|f(x/b)-a|<\varepsilon$. This means $$\limti x g(x)=a.$$

This can be viewed as a special case of the theorem about limit of composite function, since $x/b\to\infty$ for $x\to\infty$.

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