Q. 23 Find the roots, discontinuous, a... [FREE SOLUTION]

Chapter 1: Q. 23 (page 149)

Find the roots, discontinuous, and horizontal and vertical asymptotes of the function in Exercises 23-24. Support your answers by explicitly computing any relevant limits.
$f (x) = \frac{x^{2} - 2 x - 3}{x - 3}$

Short Answer

Expert verified

The root of $f (x)$ is $0 .$ $f (x)$ discontinuous at $x = 3 .$ $f (x)$ does not have any horizontal asymptotes. $f (x)$ has vertical asymptotes at $x = 3 .$

Step by step solution

Step 1. Given information

The given function is $f (x) = \frac{x^{2} - 2 x - 3}{x - 3} .$ We need to determine the roots, discontinuous, and horizontal, and vertical asymptotes of the function $f (x) .$

Step 2. Solution

$f (x) = \frac{x^{2} - 2 x - 3}{x - 3} .$

$= \frac{x^{2} - 3 x + x - 3}{x - 3} .$

$= \frac{x (x - 3) + 1 (x - 3)}{(x - 3)} .$

localid="1649221655200" $= \frac{(x + 1) (x - 3)}{(x - 3)} .$

The root of the function is $x = - 1$ and the $f (x)$ is discontinuous at $x = 3 .$ To determine the horizontal asymptotes we must examine limit as $x 鈫� 卤鈭� .$

localid="1648209600492" $\lim_{x 鈫� 鈭�} f (x) = \frac{(x - 3) (x + 1)}{(x - 3)} .$

The function $f (x)$ does not have any horizontal asymptote.

The value of $x$ that cause the denominator of $f (x)$ to zero is $3 .$

So, $\lim_{x 鈫� 鈭�} f (x) = \frac{(x - 3) (x + 1)}{(x - 3)} .$

$\lim_{x 鈫� 3} f (x) = x + 1 .$

$= 4 .$

So, the vertical asymptotes at $x = 3 .$

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91影视

Find the roots, discontinuous, and horizontal and vertical asymptotes of the function in Exercises 23-24. Support your answers by explicitly computing any relevant limits.
$f (x) = \frac{x^{2} - 2 x - 3}{x - 3}$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solution

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91影视

Find the roots, discontinuous, and horizontal and vertical asymptotes of the function in Exercises 23-24. Support your answers by explicitly computing any relevant limits.fx=x2-2x-3x-3

Short Answer

Step by step solution

Step 1. Given information

Step 2. Solution

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Calculus

Decision Maths

Discrete Mathematics

Theoretical and Mathematical Physics

Applied Mathematics

Logic and Functions

Study anywhere. Anytime. Across all devices.

Find the roots, discontinuous, and horizontal and vertical asymptotes of the function in Exercises 23-24. Support your answers by explicitly computing any relevant limits.
$f (x) = \frac{x^{2} - 2 x - 3}{x - 3}$