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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 $\sqrt{1+x^2}/x$? – David Mitra Aug 22 '12 at 22:47
Maybe this thread is of interest to you. – t.b. Aug 22 '12 at 22:48
@AsafKaragila Fixed! :) – Korgan Rivera Aug 22 '12 at 22:56
@DavidMitra Ha! That example is perfect. – Korgan Rivera Aug 22 '12 at 23:01
@t.b. Thanks, that's exactly the sort of thing I wanted to see. – Korgan Rivera Aug 22 '12 at 23:04

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