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Limits of a Polynomial/Rational Function

  1. Sep 24, 2005 #1
    I understand that the limit as x -> a for a polynomial function, f(x), is equal to f(a) because the function is always continuous.

    However, how can I prove this?

    I also have to prove this for a rational function, including the fact that the denominator cannot equal to 0.

    Thank you.
     
  2. jcsd
  3. Sep 24, 2005 #2
    Ummm, you could try using the definition of a limit maybe:

    [tex]\left|f(x)-L\right|<\epsilon[/tex]

    [tex]0<\left|x-a\right|<\delta[/tex]
     
  4. Sep 24, 2005 #3
    What lemmas do you have available? I.e., do you know how to prove that the sum/product/quotient of two continuous functions is continuous?
     
  5. Sep 24, 2005 #4

    HallsofIvy

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    By the time you get to the definition of "continuous function", you should already know:

    1) [tex]lim_{x->a} c= c[/tex] where c is a constant

    2) [tex]lim_{x->a} x= a[/tex]

    3) If [tex]lim_{x->a}f(x)= L[/tex] and [tex]lim_{x->a}g(x)= M[/tex] then
    [tex]lim_{x->a}f(x)+ g(x)= L + M[/tex]

    4) If [tex]lim_{x->a}f(x)= L[/tex] and [tex]lim_{x->a}g(x)= M[/tex] then
    [tex]lim_{x->a}f(x)g(x)= LM[/tex]

    Since all polynomials consist of sums of products of x with itself and constants, it should be easy to use those to prove that [tex]lim_{x->a}P(x)= P(a)[/tex] for P(x) any polynomial.
     
  6. Sep 28, 2005 #5
    Can you at least give me the first step?

    I asked my professor and he also said to use the limit laws, but how can I begin?

    Once I know the beginning I think I can figure out the rest on my own.

    Thank you.
     
  7. Sep 28, 2005 #6

    CRGreathouse

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    Decompose the polynomial into a collection of operations known to preserve continuity.
     
  8. Sep 28, 2005 #7
    Thank you. That has helped.
     
  9. Sep 29, 2005 #8

    HallsofIvy

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    To show that a rational function is continuous every where except where the denominator is 0, you will need one more limit property:

    If [tex]lim_{x->a}f(x)= L[/tex] and [tex]lim_{x->a}g(x)= M[/tex] then [tex]lim_{x->a}\frac{f(x)}{g(x)}= \frac{L}{M}[/tex] provided M is not 0.
     
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