Limits of a Polynomial/Rational Function

  • Thread starter dekoi
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  • #1
dekoi

Main Question or Discussion Point

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.
 

Answers and Replies

  • #2
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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]
 
  • #3
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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?
 
  • #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.
 
  • #5
dekoi
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.
 
  • #6
CRGreathouse
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dekoi said:
Can you at least give me the first step?
Decompose the polynomial into a collection of operations known to preserve continuity.
 
  • #7
dekoi
Thank you. That has helped.
 
  • #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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