explain how l'hopital rule works and give examples of the four ways it can be used ie 0/0, inf/inf, 0/inf and inf/0?
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ok so whenever we have liimit we try to evaluaate it using the limit give itself. and if we get any of those expressions above that you wrote, we use l'hopitals rule.
so one example is lim as n->0 for sin(x) /x this would obviously reveal that it would be 0 /0 right? because sin(0) = 0
so we use lhopitals rule where we derive both the bottom and top equations
so we will get
lim as n -> 0 for cos(x)/1
where cos(x) is derivative of sin(x) and 1 is derivative of x
so then lim as n->0 of cos(x) is cos(0) = 1 which is the same answer as
lim n->0 sin(x) /x
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okidokes so whats an example of inf/inf? please :)
example is lim n -> infinity of ln(x) /x in this case both our functions ln(X) and x go to infinity. right?
so if we derive ln(x) we get (1/x) and x will become 1
so now we get lim as n-> infinity of 1/x = 0 because x will just go to infinity
so I should write what l'hopital rule is then give eg above and eg of inf/inf and say it doesn't work for inf/0 or 0/inf and give egs of how it doesn't wrk?
if it's not too much trouble fancy giving me egs of inf/0 and 0/inf?
was thinking n->inf e^x/? for inf/0 and vice versa?
or might just say they don't work? but why don't they work?
Suppose that lim(f(x)) and lim(g(x)) are both zero or both ±infinity. Then if lim(f'(x)/(g'(x))) has a finite limit or the limit is ±infinity, then lim(f(x)/(g(x))) = lim(f'(x)/(g'(x))).
so we cannot use it for inf/0 or 0 / inf