   Chapter 2.3, Problem 58E

Chapter
Section
Textbook Problem

If r is a rational function. use Exercise 57 to show that limx→a r(x) = r(a) for every number a in the domain of r.

To determine

To show: If r is a rational function, then limxar(x)=r(a) for every number a in the domain of r.

Explanation

From exercise 57, if p is a polynomial, then limxap(x)=p(a).

Proof:

Let r be a rational function.

By definition of rational function, r(x)=p(x)q(x) where p(x) and q(x) are polynomials in x and q(x) is not the zero polynomial.

Let p(x)=α0+α1x+α2x2++αnxn and q(x)=β0+β1x+β2x2++βnxn where αi,βi:i{1,2,,n} are constants.

The limit of the polynomial r(x) as x approaches a.

limxar(x)=limxap(x)q(x)

Apply limit law 5, since the denominator q(x) not equal to zero

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