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Math Help - composition of limits as x tends to infinity

  1. #1
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    composition of limits as x tends to infinity

    Let f,g: R->R be functions. Suppose f(x)->k as x-> infinity and that g is continuous at k. Prove g(f(x))->g(k) as x -> inifnity

    My attempt:

    Assume not

    Then |g(f(x)) - g(k)| > epsilon > 0 for all x > 0
    so |g(k) - g(k)| > epsilon > 0 by continuity

    which is a contradiction

    Is this correct? Or am I missing something?

    many thanks
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  2. #2
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    From the continuity of g at k we get \varepsilon  > 0\; \Rightarrow \;\left( {\exists \delta  > 0} \right)\left[ {\left| {y - k} \right| < \delta \, \Rightarrow \,\left| {g(y) - k} \right| < \varepsilon } \right].

    From the given we also have \left( {\exists N} \right)\left[ {x > N\, \Rightarrow \,\left| {f(x) - k} \right| < \delta } \right].

    Put those two together to finish.
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