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I have two functions, $f(x)$ and $g(x)$ that are continuous on the real number line. I'm taking the limit of their product:

$$\lim_{x\rightarrow\infty}f(x)g(x)$$

Suppose that the limit $\lim_{x\rightarrow\infty}g(x)=c$.

Is there a continuous function $f(x)$ such that:

$$\lim_{x\rightarrow\infty}f(x)g(x)\neq\lim_{x\rightarrow\infty}cf(x)$$

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4  
How about $g(x)=1/x$ whan $x>1$ and $g(x)=1$ otherwise, and $f(x)=x$? –  kneidell Dec 21 '11 at 21:04
2  
@kneidell You could post this as an answer, so that the OP could choose your answer and there will be one less question for math.SE –  user21436 Dec 21 '11 at 21:12

2 Answers 2

up vote 3 down vote accepted

Same goes for the following:

If $$g=\begin{cases}1/x&\text{ if }x>1\\1&\text{ otherwise}\end{cases}\text{ and }f(x)=x$$ Then both functions are continuous on the real line, $\lim_{x\to\infty}g(x)=0$, and $f(x)g(x)$ is the constant one function on the ray $[1,\infty)$, suggesting that $$\lim_{x\to\infty}f(x)g(x)=1\neq 0\cdot\lim_{x\to\infty}f(x)$$

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If $f(x)=x$ and $\lim\limits_{x\to\infty}g(x)=c$, and $c$ is a real number (thus not $\infty$ or $-\infty$), then $f$ is continuous and $\lim\limits_{x\to\infty}\Big(f(x)g(x)\Big)$ does not exist.

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I think the author should edit his TeX to fix the glitch in viewing after the edit I posted was rejected by @mixedmath as being too minor. –  user21436 Dec 21 '11 at 21:33
    
@Kannappan: I see the glitch too, but it depends on your screen width. So your edit may have made things worse for many users. –  TonyK Dec 21 '11 at 23:40
    
I think in this case neither $\lim_{x\rightarrow\infty}f(x)g(x)$ nor $\lim_{x\rightarrow\infty}cf(x)$ exists. Not sure if it's correct to say $\lim_{x\rightarrow\infty}f(x)g(x)=\lim_{x\rightarrow\infty}cf(x)$ though... –  M.B.M. Dec 22 '11 at 1:13
    
What is the glitch? I don't see anything abnormal. If your edit is rejected because of not enough characters, you can add a comment at the bottom that it was rejected because of that, and with the comment there are enough characters. –  Michael Hardy Dec 22 '11 at 1:33
    
@MichaelHardy I do not see any glitch in your TeX... –  M.B.M. Dec 22 '11 at 1:37

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