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Math Help - Prove a limit of 2 functions multiplied is 0

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    Prove a limit of 2 functions multiplied is 0

    R= Reals
    Let f, g be defined on A as a subset of R to R, and let c be a cluster point of A. Suppose that f is bounded on a neighborhood of c and that the limit as
    x --> c g=0. Prove that the limit as x --> c fg=0
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    Super Member redsoxfan325's Avatar
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    Quote Originally Posted by gixxer998 View Post
    R= Reals
    Let f, g be defined on A as a subset of R to R, and let c be a cluster point of A. Suppose that f is bounded on a neighborhood of c and that the limit as
    x --> c g=0. Prove that the limit as x --> c fg=0
    If f is bounded on a neighborhood of c, then -M\leq f(x)\leq M for some M\in\mathbb{R}. So \lim_{x\to c}-Mg(x)\leq \lim_{x\to c}f(x)g(x)\leq \lim_{x\to c}Mg(x)

    Proceed using properties of limits.
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  3. #3
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    Quote Originally Posted by gixxer998 View Post
    R= Reals
    Let f, g be defined on A as a subset of R to R, and let c be a cluster point of A. Suppose that f is bounded on a neighborhood of c and that the limit as
    x --> c g=0. Prove that the limit as x --> c fg=0
    Hint: |fg(x)-0|=|f(x)g(x)|\le B|g(x)|
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