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Math Help - Limit Problem

  1. #1
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    Limit Problem

    Examples of functions satisfying each of the following conditions

    (a) lim as x goes to zero of f of x^2 exists, but the lim as x goes to zero of f of x does not.

    (b) lim as x goes to zero of f of 1/x exist, but the lim as x goes to zero of f of x does not.

    (c) f(x) is not continuous at 1, but g(x) = |f(x)| is continuous at 1.

    (d) f(x) is nto continuous anywhere, but g(x) = |f(x)| is continuous everywhere.

    For each condition provide one function.
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  2. #2
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    Re: Limit Problem

    Quote Originally Posted by calcmaster View Post
    Examples of functions satisfying each of the following conditions

    (a) lim as x goes to zero of f of x^2 exists, but the lim as x goes to zero of f of x does not.

    (b) lim as x goes to zero of f of 1/x exist, but the lim as x goes to zero of f of x does not.

    (c) f(x) is not continuous at 1, but g(x) = |f(x)| is continuous at 1.

    (d) f(x) is nto continuous anywhere, but g(x) = |f(x)| is continuous everywhere.
    Here are two functions for you to think about.

    f(x) = \left\{ {\begin{array}{rl}  {1,}&{x \geqslant 0} \\   { - 1,}&{x < 0} \end{array}} \right. AND g(x)=\frac{1}{x}\text{ if }x\ne 0,~\&~0\text{ otherwise}.
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  3. #3
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    Re: Limit Problem

    Another interesting function would be f(x)= 1 if x is irrational, f(x)= -1 if x is rational.
    Last edited by HallsofIvy; July 5th 2012 at 12:12 PM.
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