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Thread: Real Analysis, Limit of a Function

  1. #1
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    Real Analysis, Limit of a Function

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


    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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  2. #2
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    Quote Originally Posted by jstew View Post
    Prove that $\displaystyle \lim_{x\to c}f(x) =L$ if and only if $\displaystyle \lim_{x\to 0}f(x+c) =L$.
    From the given we know, $\displaystyle \varepsilon > 0\, \Rightarrow \,\left( {\exists \delta > 0} \right)\left[ {0 < \left| {y - c} \right| < \delta \, \Rightarrow \,\left| {f(y) - L} \right| < \varepsilon } \right]$.

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