Chapter 13, Problem 9RCC

a.

To determine

To state: the meaning of that the line $y=L$ is a horizontal asymptote of the curve $y=f\left(x\right)$.

Answer to Problem 9RCC

the meaning of that the line $y=L$ is a horizontal asymptote of the curve $y=f\left(x\right)$ is that $\underset{x\to \infty }{\lim }f\left(x\right)=L\text{ or }\underset{x\to -\infty }{\lim }f\left(x\right)=L$.

Explanation of Solution

Given information:

The average rate of change of $y$ with respect to $x$ between the numbers $a$ and $x$.

the meaning of that the line $y=L$ is a horizontal asymptote of the curve $y=f\left(x\right)$ is that $\underset{x\to \infty }{\lim }f\left(x\right)=L\text{ or }\underset{x\to -\infty }{\lim }f\left(x\right)=L$.

  

b.

To determine

To find: that which curves have horizontal asymptotes.

Answer to Problem 9RCC

the curves which have horizontal asymptotes are $y={\tan }^{-1}x$ , $y=\frac{1}{x}$ , $y=e^x$.

Explanation of Solution

Given information:

The curves are:

  $y=x^2$

  $y={\tan }^{-1}x$

  $y=\frac{1}{x}$

  $y=e^x$

  $y=\sin x$

  $y=\ln x$

Calculation: form the above curves $y={\tan }^{-1}x$ , $y=\frac{1}{x}$ , $y=e^x$ have horizontal asymptote, because these functions have $\underset{x\to \infty }{\lim }f\left(x\right)=L$.