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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