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Simple question about limits

  1. Mar 20, 2009 #1
    If f is continuous in some neighborhood of x = a, then is the following true:

    [tex] \lim_{x \rightarrow a} f(x) = f( \lim_{x \rightarrow a} x) [/tex]?
     
  2. jcsd
  3. Mar 20, 2009 #2
    If f is continuous in some neighborhood of x = a, it is also continuous at x = a, because x = a is contained in the neighborhood. The l.h.s equals the r.h.s because of the fact that the limit as x tends to 'a' of f(x) equals 'f(a)' (because of continuity of f) and on the other hand, f of the limit of x as 'x tends to a' is obviously f(a) since lim(x) = a as x --> a
     
    Last edited: Mar 20, 2009
  4. Mar 20, 2009 #3
    This an anorthodoxe way of writting : f is continuous at x=a <====>
    [tex] \lim_{x \rightarrow a} f(x)=f(a)[/tex]

    But i suppose is correct since [tex]\lim_{x\rightarrow a}x = a[/tex]
     
  5. Mar 21, 2009 #4

    lurflurf

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    Homework Helper

    That is true. This makes clear the idea of continuity.
     
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