Limits of a Polynomial/Rational Function

In summary, to prove that the limit of a polynomial function, f(x), as x approaches a is equal to f(a), you can use the definition of a limit and the limit laws for continuous functions. To prove this for a rational function, you will also need the additional limit property stated above.
  • #1
dekoi
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.
 
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  • #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]
 
  • #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?
 
  • #4
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
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
dekoi said:
Can you at least give me the first step?

Decompose the polynomial into a collection of operations known to preserve continuity.
 
  • #7
Thank you. That has helped.
 
  • #8
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.
 

1. What is a polynomial function?

A polynomial function is a mathematical expression that contains variables, coefficients, and exponents, and is constructed using only the operations of addition, subtraction, and multiplication. Examples of polynomial functions include f(x) = 3x^2 + 5x + 2 and g(x) = 2x^3 - 4x^2 + 9x - 6.

2. What is the degree of a polynomial function?

The degree of a polynomial function is the highest exponent in the expression. For example, the polynomial function f(x) = 3x^2 + 5x + 2 has a degree of 2, while the function g(x) = 2x^3 - 4x^2 + 9x - 6 has a degree of 3.

3. What are the limits of a polynomial function?

The limits of a polynomial function refer to the values that the function approaches as the input (x) approaches a certain value. This can be determined by evaluating the function at the given value, or by using algebraic methods such as factoring and canceling out common factors.

4. How do you find the horizontal asymptotes of a rational function?

To find the horizontal asymptotes of a rational function, you need to compare the degrees of the numerator and denominator. If the degree of the numerator is greater than the degree of the denominator, there is no horizontal asymptote. If the degree of the numerator is equal to the degree of the denominator, the horizontal asymptote is the ratio of the leading coefficients. If the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0.

5. How do you determine the end behavior of a polynomial function?

The end behavior of a polynomial function refers to the behavior of the function as the input (x) approaches positive or negative infinity. This can be determined by looking at the leading term of the polynomial function. If the leading term has an even exponent, the end behavior is the same on both sides of the graph. If the leading term has an odd exponent, the end behavior is opposite on each side of the graph.

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