Chapter 13.4, Problem 1E

Let f be a function defined on some interval (a, ∞). Then

$\underset{x\to \infty }{\lim }f(x)=L$

means that the values of f(x) can be made arbitrarily close to ______ by taking _______ sufficiently large. In this case the line y = L is called a _______ _______ of the function y = f(x). For example, $\underset{x\to \infty }{\lim }\frac{1}{x}=\_\_\_\_\_\_$, and the line y = _______ is a horizontal asymptote.

To determine

To fill: The blanks in the statement “Let f be a function defined on some interval $\left(a,\infty \right)$. Then $\underset{x\to \infty }{\lim }f\left(x\right)=L$ means that the values of $f\left(x\right)$ can be made arbitrarily close to _______ by taking _________ sufficiently large. In this case the line $y=L$ is called a ______ of the function $y=f\left(x\right)$. For example, $\underset{x\to \infty }{\lim }\frac{1}{x}=$_______, and the line $y=$______ is a horizontal asymptote.”

Answer to Problem 1E

Let f be a function defined on some interval $\left(a,\infty \right)$. Then $\underset{x\to \infty }{\lim }f\left(x\right)$ is $L$ means that the values of $f\left(x\right)$ can be made arbitrarily close to L by taking x sufficiently large. In this case the line $y=L$ is called a horizontal asymptote of the function $y=f\left(x\right)$. For example, $\underset{x\to \infty }{\lim }\frac{1}{x}=$0, and the line $y=$0 is a horizontal asymptote.

Explanation of Solution

Definition used:

Horizontal asymptote:

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

Let f be the function defined on the interval $\left(a,\infty \right)$.

The definition of limit $\underset{x\to \infty }{\lim }f\left(x\right)=L$, implies that $f\left(x\right)$ can be made approximately close to L when the value of x is sufficiently large.

From the definition stated above, the line $y=L$ is called horizontal asymptote of $y=f\left(x\right)$ if $\underset{x\to \infty }{\lim }f\left(x\right)=L$.

Now, consider $\underset{x\to \infty }{\lim }\frac{1}{x}$.

In $\underset{x\to \infty }{\lim }\frac{1}{x}$, as the value of x becomes larger and approaches to $\infty$, the value of $\frac{1}{x}$ approaches 0.

So, the value of $\underset{x\to \infty }{\lim }\frac{1}{x}=0$.

Thus, from the definition stated above, the line $y=0$ is the horizontal asymptote of the function $y=\frac{1}{x}$.