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I was reading that, when trying to solve something like:

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

I can rewrite is as:

$$\lim_{x\to\infty} \frac{f(x)}{\frac{1}{g(x)}}$$

and use L'Hospital's Rule to solve. And, if this doesn't work, I can try using the other function as the denominator:

$$\lim_{x\to\infty} \frac{g(x)}{\frac{1}{f(x)}}$$

So I wondered: are there well-known quotients of functions that don't work in either case and, if so, how do I then solve them?

An example that doesn't submit to this process is:

$$\lim_{x\to\infty} x.x$$

But obviously L'Hospital's Rule would not be necessary in this case.

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    Something like $\sqrt{1+x^2}/x$?2012-08-22
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    Maybe [this thread](http://math.stackexchange.com/q/59842/5363) is of interest to you.2012-08-22
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    @AsafKaragila Fixed! :)2012-08-22
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    @DavidMitra Ha! That example is perfect.2012-08-22
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    @t.b. Thanks, that's exactly the sort of thing I wanted to see.2012-08-22

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