Original post by Salimshady007
If lim x->2 ((f(x)-5)/(x-2))=3; find lim x->2 (f(x)) ?
If find
Well, I guess we could hand wave a bit and say that if x is close to 2, say, then the fractional expression must be close to 3, say, , where gets smaller as gets smaller. Then rearrange things and find the required limit.
Or else, I guess we could argue like this: suppose the numerator tends to a number, , say. Then what happens to the fraction as the denominator gets smaller?
Original post by atsruser
Well, I guess we could hand wave a bit and say that if x is close to 2, say, then the fractional expression must be close to 3, say, , where gets smaller as gets smaller. Then rearrange things and find the required limit.Assuming algebra of limits has been covered, the simplest thing is to multiply by a suitable function g(x) and apply said limit rules...
Original post by DFranklin
Assuming algebra of limits has been covered, the simplest thing is to multiply by a suitable function g(x) and apply said limit rules...
Yep, good idea. It'd help with some of these questions to have a little more background info: A level, uni, maths degree, engineering, maths for bio/geo/psych-ologists ...?
Original post by atsruser
Yep, good idea. It'd help with some of these questions to have a little more background info: A level, uni, maths degree, engineering, maths for bio/geo/psych-ologists ...?
I mean, for a question like this, I think it's important that the OP is aware of how to do it using the approach you suggest (because it's more instructive about how to think about limits), but in an exam, algebra of limits kills this question in a couple of lines.
Edit: for an example from a different area, I found the idea that you can inject arbitrary sequences of +ve integers into by the map gave very short and easy solutions to a lot of countability problems. But I'm not sure it actually gave you a good background for thinking about more general injection/bijection/etc problems.
(edited 6 years ago)
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