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Thread: Limit definition in Rn

  1. #1
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    Limit definition in Rn

    I have a question.
    Let f : Rn → Rm and let l ∈ Rm. Show that lim x→c f(x) = l ⇔ lim x→c||f(x)−l|| = 0.

    How can I do this? I know the (ε, δ)-definition of limit, but this is in Rn.

    Thank you
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  2. #2
    MHF Contributor
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    Re: Limit definition in Rn

    You could write down the definition of the left hand limit, and then define a scalar function g: \mathbb R_n \mapsto \mathbb R defined by g(\mathbf{x}) = \big| |\mathbf{f}(\mathbf{x}) - \mathbf{l}| \big| and show that the (\epsilon, \delta) definition of the limit
    \lim_{\mathbf{x} \to \mathbf{c}} g(\mathbf{x}) = 0

    is satisfied.

    I think you can do the same thing in reverse to get the other half of the proof.
    Thanks from Cyn1
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