   Chapter 9.2, Problem 37E Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340

Solutions

Chapter
Section Mathematical Applications for the ...

12th Edition
Ronald J. Harshbarger + 1 other
ISBN: 9781337625340
Textbook Problem

For Problems 37 and 38, let f ( x ) = a n x n + a n − 1 x n − 1 + … + a 1 x + a 0 b m x m + b m − 1 x m − 1 + … + b 1 x + b 0 be a rational functionIf m = n, show that lim x → ∞ f ( x ) = a n b n and hence that y = a n b n is a horizontal asymptote.

To determine

To prove: The value of the limit, limxf(x)=anbn, if m=n, and that y=anbn is a horizontal asymptote.

Explanation

Given Information:

The value of m is equal to n. The provided rational function is f(x)=anxn+an1xn1++a1x+a0bmxm+bm1xm1++b1x+b0.

Formula used:

A limit limxcx can be simplified as,

limxcx=c

Calculation:

Consider the provided rational function,

f(x)=anxn+an1xn1++a1x+a0bmxm+bm1xm1++b1x+b0

Consider the limit at x,

limxanxn+an1xn1++a1x+a0bmxm+bm1xm1++b1x+b0

Take xn common from the numerator and xm common from the denominator,

limxanxn+an1xn1++a1x+a0bmxm+bm1xm1++b1x+b0=limxan+an1xxn++a1xn1xn+a0xnxnbn+bm1xxm++b1xm1xm+b0xmxm=limxxn(an+an1x++a1xn1+a0

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