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Math Help - How to solve this limit?

  1. #1
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    How to solve this limit?

    Yeah so I'm really rusty with the basic calc stuff. Pretty much forgot how to do this.

    Evaluate:

    lim
    x -> infinity x sin (1/x)


    So from the limit laws we know that lim x -> a f(x) g(x) = lim x->a f(x) lim x-> a g(x)
    However thats assuming both function's limits exist.

    If I let f(x) = x and g(x) = sin (1/x), the limit of f(x) is just infinity while g(x) is 0.

    Using arbitrarily large numbers I can see that the answer is 1 but I don't know how to prove it or get there in a proper way.
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  2. #2
    MHF Contributor

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    Re: How to solve this limit?

    Quote Originally Posted by Kuma View Post
    Yeah so I'm really rusty with the basic calc stuff. Pretty much forgot how to do this.

    Evaluate:

    lim
    x -> infinity x sin (1/x)


    So from the limit laws we know that lim x -> a f(x) g(x) = lim x->a f(x) lim x-> a g(x)
    However thats assuming both function's limits exist.

    If I let f(x) = x and g(x) = sin (1/x), the limit of f(x) is just infinity while g(x) is 0.

    Using arbitrarily large numbers I can see that the answer is 1 but I don't know how to prove it or get there in a proper way.
    Let y= 1/x. Then the problem becomes \lim_{y\to 0}\frac{sin(y)}{y} which is a well known limit.
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  3. #3
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    Re: How to solve this limit?

    Quote Originally Posted by Kuma View Post
    Yeah so I'm really rusty with the basic calc stuff. Pretty much forgot how to do this.

    Evaluate:

    lim
    x -> infinity x sin (1/x)


    So from the limit laws we know that lim x -> a f(x) g(x) = lim x->a f(x) lim x-> a g(x)
    However thats assuming both function's limits exist.

    If I let f(x) = x and g(x) = sin (1/x), the limit of f(x) is just infinity while g(x) is 0.

    Using arbitrarily large numbers I can see that the answer is 1 but I don't know how to prove it or get there in a proper way.
    This question has been asked many times in these forums, most recently here: http://www.mathhelpforum.com/math-he...ts-188132.html

    Thread closed.
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