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Consider the limit product rule: $$\lim_{x\rightarrow c} (f(x)⋅g(x))=[\lim _{x\rightarrow c} f(x)]⋅[\lim_{x\rightarrow c} g(x)]$$

Now consider, for the sake of the argument, $f(x) = x, g(x) = (e/x)$

Clearly, the limit is e. However, by the product it would be impossible to figure out. Does this mean that the product rule is only valid when the components don't have a limit of 0 or infinity? Will it always work for other cases?

Ashish Gaurav
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1 Answers1

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Usually such a rule would be stated:

"If $\lim_{x\to c} f(x)$ and $\lim_{x \to c} g(x)$ both exist, then $\lim_{x \to c} f(x)g(x) = \lim_{x \to c} f(x) \lim_{x \to c} g(x)$"

So in your supplied example, one of the limits does not exist and the rule would not apply. For that case you would examine $fg$ instead of trying to use algebra of limits.

Jason Knapp
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  • Actually a slight generalization is used in practice. Let $\lim_{x \to 0}g(x)$ exist and be non-zero. Then $\lim_{x \to 0}f(x)g(x)$ exists (or does not exist) if $\lim_{x \to 0}f(x)$ exists (or does not exist). Thus the behavior of $f(x)g(x)$ is similar to that of $f(x)$ as far existence of limit is concerned. – Paramanand Singh Dec 20 '14 at 06:51