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Thread: limit proof

  1. #1
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    limit proof

    I'm sure the proof is really easy, but I am relatively new to these proofs and I'm not really sure how to start...

    Let f: R $\displaystyle \rightarrow$ R and let c $\displaystyle \in$ R. Show lim as x $\displaystyle \rightarrow$ c of f(x) = L if and only if lim x $\displaystyle \rightarrow$ 0 of f(x+c) = L

    (sorry about the code, this is the first time I have used Latex...can anyone suggest an all in one reference for Latex coding?)
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  2. #2
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    Quote Originally Posted by davido View Post
    Let f: R $\displaystyle \rightarrow$ R and let c $\displaystyle \in$ R. Show lim as x $\displaystyle \rightarrow$ c of f(x) = L if and only if lim x $\displaystyle \rightarrow$ 0 of f(x+c) = L
    From the given $\displaystyle \left| {y - c} \right| < \delta \Rightarrow \quad \left| {f(y) - L} \right| < \varepsilon $.
    Let $\displaystyle (x+c)=y$.
    $\displaystyle \left| x \right| < \delta \Rightarrow \quad \left| {y - c} \right| = \left| {\left( {x + c} \right) - c} \right| = \left| x \right| < \delta \Rightarrow \quad \left| {f(x + c) - L} \right| = \left| {f(y) - L} \right| < \varepsilon $
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