How do I handle 0 times infinity when using L'Hospital's rule?

[HannahQS]

I was working through a problem about the convergence of a series and ended up with an expression that simplifies to zero times infinity. My textbook just says this is an indeterminate form that requires further analysis, but I’m stuck on how to actually approach rewriting it. I feel like I’m missing a step in setting up the limit properly to apply L’Hôpital’s rule.

[Richard92]

I have run into the 0 times infinity thing too. The trick is to flip the product into a ratio so the limit can be tackled with a rule like L'Hôpital. For example with a limit like x times log x as x goes to zero from the right you can rewrite as log x divided by 1 over x. Then as x goes to zero from the right log x goes to minus infinity and 1 over x goes to plus infinity and the ratio is a valid infinity over infinity form. Differentiating top and bottom gives one over x over minus one over x squared which simplifies to minus x which goes to zero. The catch is to keep track of the signs and the domain and to verify the rewriting is legitimate before applying the rule.

[AddisonUM]

I tried the same idea by turning the product into f of x over h of x when h goes to zero. The plan is to differentiate top and bottom so you get a new limit and hope it exists. But you have to check that the functions are differentiable in a neighborhood and that the limit of the derivatives exists.

[Sofia85]

Are you sure this is the real snag or is the issue something else in the setup? Sometimes the indeterminate comes from a step that is not the core, and chasing the form hides what is actually unsettled about the series or the limit you are trying to control.

[Timothy_H]

I tried a quick approach and got stuck fast. I wrote it as a ratio and did one derivative and the bottom vanished or the rates refused to settle. I ended up abandoning that path and went back to the drawing board.