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Math Help - Real Analysis, Limit of a Function

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    Real Analysis, Limit of a Function

    Prove that \lim_{x\to c}f(x) =L if and only if \lim_{x\to 0}f(x+c) =L.


    I know that I'm to use the epsilon-delta definition of a limit, but for some reason I just can't see how to go about this.
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    Quote Originally Posted by jstew View Post
    Prove that \lim_{x\to c}f(x) =L if and only if \lim_{x\to 0}f(x+c) =L.
    From the given we know, \varepsilon  > 0\, \Rightarrow \,\left( {\exists \delta  > 0} \right)\left[ {0 < \left| {y - c} \right| < \delta \, \Rightarrow \,\left| {f(y) - L} \right| < \varepsilon } \right].

    Now observe: 0<\left| x \right| < \delta \, \Rightarrow \,0,\left| {\left( {x + c} \right) - c} \right| < \delta \, \Rightarrow \,\varepsilon  > 0\, \Rightarrow \,\left[ {\,\left| {f(x + c) - L} \right| < \varepsilon } \right]
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