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Homework Help: Limits - Apostol book

  1. Mar 18, 2013 #1
    Hello guys, I am stuck in page 129 on Calculus Vol I - Apostol book. I would like to know if there is anybody here who can help me. I am not a mathematician, so It might be a simple transformation but I am not going through it.

    He states that:

    lim(x->p) f(x) = A is equivalent to say that:

    lim(x->p) ( f(x) - A) = 0 OR

    lim(x->p) | f(x) - A| = 0

    I can not make these transformations algebraically, how can it be done?

    Thanks for any help
  2. jcsd
  3. Mar 18, 2013 #2


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    Would you agree a=b, a-b=0 and |a-b|=0 all mean the same thing? That's really almost the whole story. There is nothing complicated going on here.
    Last edited: Mar 18, 2013
  4. Mar 18, 2013 #3
    The book's definition of a limit is such that:

    | f(x) -A | < ε whenever 0 < | x - p | < δ

    So it presents the three equations (on the previous post) and it says: The equivalence becomes apparent as soon as we write each of these statements in the ε, δ terminology.

    That is the core of my doubts. How to write those three equations.

    Sorry that I wasn't clear enough in the first post.
  5. Mar 18, 2013 #4


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    The definition of lim(x->p) f(x)=A is for all ε>0 there exists a δ>0 such that |f(x)-A|<ε whenever 0<|x-p|< δ. Don't leave out the quantifiers. Now follow their advice and write out what lim(x->p) (f(x)-A)=0 means.
  6. Mar 18, 2013 #5
    that is exactly the point where I am confused.

    Since we have:
    lim(x->p) f(x) = A, then, in my opinion, we would have

    lim(x->p) ( f(x) - A) = 0, since A is defined in the line above:

    lim(x->p) ( f(x) - lim(x->p) f(x) ) = 0

    And I don't know how to go any further from here. ;/

    Alternatively, iff you consider:
    lim(x->p) ( f(x) -A ) = 0, as we know |f(x) -A| < ε
    lim(x->p) ( ε ) would be ε, not zero.
    Last edited: Mar 19, 2013
  7. Mar 19, 2013 #6


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    If lim(x->p) g(x)=0 then what's the definition of that? Spell it out for me. Then put g(x)=f(x)-A.
  8. Mar 19, 2013 #7
    Thank you, that was good.
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