Chapter 4.3, Problem 57AYU

Graph each of the following functions:

$y=\frac{x^2-1}{x-1}y=\frac{x^3-1}{x-1}$

$y=\frac{x^4-1}{x-1}y=\frac{x^5-1}{x-1}$

Is $x=1$ a vertical asymptote? Why not? What is happening for $x=1$?

What do you conjecture about $y=\frac{x^n-1}{x-1},n\ge 1$ an integer, for $x=1$?

To determine

To find: Is $x=\text{ 1}$ a vertical asymptote?

Why not?

What is happening for $x=\text{ 1}$?

What do you conjecture about $y=\frac{x^n-1}{x-1},n\ge 1$ an integer, for $x=\text{ 1}$?

Answer to Problem 57AYU

No, each function has a quotient of polynomials, but it is not written in lowest terms. Each function is undefined for $x=\text{ 1}$; each graph has a hole at $x=\text{ 1}$.

Explanation of Solution

Given:

$y=\frac{x^2-1}{x-1}$

No, each function has a quotient of polynomials, but it is not written in lowest terms. Each function is undefined for $x=\text{ 1}$; each graph has a hole at $x=\text{ 1}$.

To determine

To find: Is $x=\text{ 1}$ a vertical asymptote?

Why not?

What is happening for $x=\text{ 1}$?

What do you conjecture about $y=\frac{x^n-1}{x-1},n\ge 1$ an integer, for $x=\text{ 1}$?

Answer to Problem 57AYU

No, each function has a quotient of polynomials, but it is not written in lowest terms. Each function is undefined for $x=\text{ 1}$; each graph has a hole at $x=\text{ 1}$.

Explanation of Solution

Given:

$y=\frac{x^3-1}{x-1}$

No, each function has a quotient of polynomials, but it is not written in lowest terms. Each function is undefined for $x=\text{ 1}$; each graph has a hole at $x=\text{ 1}$.

To determine

To find: Is $x=\text{ 1}$ a vertical asymptote?

Why not?

What is happening for $x=\text{ 1}$?

What do you conjecture about $y=\frac{x^n-1}{x-1},n\ge 1$ an integer, for $x=\text{ 1}$?

Answer to Problem 57AYU

No, each function has a quotient of polynomials, but it is not written in lowest terms. Each function is undefined for $x=\text{ 1}$; each graph has a hole at $x=\text{ 1}$.

Explanation of Solution

Given:

$y=\frac{x^4-1}{x-1}$

No, each function has a quotient of polynomials, but it is not written in lowest terms. Each function is undefined for $x=\text{ 1}$; each graph has a hole at $x=\text{ 1}$.

To determine

To find: Is $x=\text{ 1}$ a vertical asymptote?

Why not?

What is happening for $x=\text{ 1}$?

What do you conjecture about $y=\frac{x^n-1}{x-1},n\ge 1$ an integer, for $x=\text{ 1}$?

Answer to Problem 57AYU

No, each function has a quotient of polynomials, but it is not written in lowest terms. Each function is undefined for $x=\text{ 1}$; each graph has a hole at $x=\text{ 1}$.

Explanation of Solution

Given:

$y=\frac{x^5-1}{x-1}$

No, each function has a quotient of polynomials, but it is not written in lowest terms. Each function is undefined for $x=\text{ 1}$; each graph has a hole at $x=\text{ 1}$.