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## Main Question or Discussion Point

To deal with the indeterminate form ##0⋅\pm \infty##, we write the product ##f(x)g(x)## as ##\frac{f(x)}{1/g(x)}## or ##\frac{g(x)}{1/f(x)}##, before applying L'Hospital's rule to one of these forms.

However, on occasion, applying L'Hospital's rule to one of these forms gets us nowhere (##\lim_{x \rightarrow -\infty} x e^x## for instance), despite working for the other (equivalent) quotient.

Is there a way to know beforehand whether ##f(x)g(x)## should be expressed as ##\frac{f(x)}{1/g(x)}## or ##\frac{g(x)}{1/f(x)}##? Or is it always based on trial and error?

However, on occasion, applying L'Hospital's rule to one of these forms gets us nowhere (##\lim_{x \rightarrow -\infty} x e^x## for instance), despite working for the other (equivalent) quotient.

Is there a way to know beforehand whether ##f(x)g(x)## should be expressed as ##\frac{f(x)}{1/g(x)}## or ##\frac{g(x)}{1/f(x)}##? Or is it always based on trial and error?