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Theorem for limits?

  1. Feb 13, 2015 #1
    I read in a calculus book that. "Given ##\lim_{x \to a}\frac{f(x)}{g(x)} = c(c\neq 0)##, when ##\lim_{x \to a}g(x) = 0##, then ##\lim_{x \to a}f(x) = 0##. Why is this true?
     
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  3. Feb 13, 2015 #2

    DEvens

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  4. Feb 13, 2015 #3

    Mark44

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    A non-rigorous explanation is that, since ##\frac{f(x)}{g(x)} \to c##, where c ≠ 0, then f and g are approximately equal near a. If g approaches zero as x approaches a, then so does f.
     
  5. Feb 13, 2015 #4

    PeroK

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    Have you tried finding a counterexample? Usually a good way to see why something is true is to try to show that it's false.

    And, why must you have ##c \ne 0##?
     
  6. Feb 13, 2015 #5

    Mark44

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    Last edited by a moderator: May 7, 2017
  7. Feb 13, 2015 #6

    mathman

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    Multiply by g(x). Limit for f(x) = c(limit for g(x)) = 0.
     
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