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belliott4488 said:I don't understand how L'Hopital works on a discontinuous function such as the one you've suggested.
Well, it works the same as if the function were continuous, because it doesn't actually make any claim about the value of the ratio AT the point a, only the limit as you approach a.
It can be useful for example, if you WANT to define a continuous function but you need to know "what value should I assign at x=a that will make the function continuous?"
For example, if I want to define a function of the form sin(x)/x, this expression itself is undefined at x=0, but I may be able to continuously extend it to x=0 by defining
f(x) = sin(x)/x if x is nonzero
f(0) = c
Then L'Hopital's rule will tell me what value of c (if any) will make f continuous.