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    If lim x->2 ((f(x)-5)/(x-2))=3; find lim x->2 (f(x)) ?
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    (Original post by Salimshady007)
    If lim x->2 ((f(x)-5)/(x-2))=3; find lim x->2 (f(x)) ?
    If \displaystyle \lim_{x \to 2} \dfrac{f(x)-5}{x-2} = 3 find \displaystyle \lim_{x \to 2} f(x)

    Well, I guess we could hand wave a bit and say that if x is close to 2, say, x-2 = \delta then the fractional expression g(x) must be close to 3, say, g(x)=3 \pm \epsilon, where \epsilon gets smaller as \delta gets smaller. Then rearrange things and find the required limit.

    Or else, I guess we could argue like this: suppose the numerator f(x)-5 tends to a number, n \ne 0, say. Then what happens to the fraction as the denominator gets smaller?
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    (Original post by atsruser)
    If \displaystyle \lim_{x \to 2} \dfrac{f(x)-5}{x-2} = 3 find \displaystyle \lim_{x \to 2} f(x)

    Well, I guess we could hand wave a bit and say that if x is close to 2, say, x-2 = \delta then the fractional expression g(x) must be close to 3, say, g(x)=3 \pm \epsilon, where \epsilon gets smaller as \delta 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...
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    (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 ...?
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    (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 ...?
    But even then, you really need context like "it's a first year analysis course, we've covered all the material and generally I'm happy with it but I can't see how to do this question".

    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 \mathbb{N} by the map f(a_1, a_2, ..., a_n) = 2^{a_1} 3^{a_2} 5^{a_3} ... p_n^{a_n} 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.
 
 
 
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