Chapter 4.2, Problem 60AYU

To determine

To explain:

The degree of the numerator and denominator is same.

Answer to Problem 60AYU

The degree of the numerator and denominator is same.

Explanation of Solution

Given information:

Horizontal asymptote: $y=2$

Consider;

The function is $f\left(x\right)=\frac{a{x}^m+b{x}^{m-1}+\mathrm{...}+k}{a_1{x}^p+b_1{x}^{p-1}+\mathrm{...}+k_1}$

Where,

Constant: $k_1,k$

Assume, $m=p$ then the horizontal asymptote is $\frac{a}{a_1}$ as the degree of the denominator and numerator is same. So, multiply or divide the function by $x^m$ or $x^p$ . This is fraction or integer.

Assume, $m\ne p$ , if $m>p$ , that means there is no perfect division occurs in the function which results that $\underset{x\to \infty }{\lim }f\left(x\right)$ is infinite number.

The case $m<p$ is also the case of not perfect division occurs in the function. But the value of $\underset{x\to \infty }{\lim }f\left(x\right)=0$ .

Hence, $2$ is only horizontal asymptote when $m=p$ . In other two cases, get the results as infinite, fraction or negative integers.