**Finding Vertical Asymptotes and Holes of a Rational Function**Consider the rational function given by:\[ f(x) = \frac{x - 6}{x^2 - 36} \]**Objective:**Determine the vertical asymptotes and the x-values corresponding to any holes in the graph of the function.**Options:**- **A.** Vertical asymptote(s) at $ x = \_\_\_ $. There are no holes.- **B.** There are no vertical asymptotes, but there is/are hole(s) corresponding to $ x = \_\_\_ $.- **C.** Vertical asymptote(s) at $ x = \_\_\_ $ and hole(s) corresponding to $ x = \_\_\_ $.- **D.** There are no discontinuities.**Instructions:**- Select the correct option and fill in the x-values where necessary.- Type an integer or a fraction; use commas to separate answers as needed.**Action:**Click the answer box to enter your selection.---*Note: The question requires users to identify and classify the points of discontinuity for the given function, differentiating between vertical asymptotes and holes by analyzing the factored form of the denominator.*

Answered: **Finding Vertical Asymptotes and Holes…

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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Question

**Finding Vertical Asymptotes and Holes of a Rational Function**

Consider the rational function given by:

\[ f(x) = \frac{x - 6}{x^2 - 36} \]

**Objective:**
Determine the vertical asymptotes and the x-values corresponding to any holes in the graph of the function.

**Options:**

- **A.** Vertical asymptote(s) at $ x = \_\_\_ $. There are no holes.
- **B.** There are no vertical asymptotes, but there is/are hole(s) corresponding to $ x = \_\_\_ $.
- **C.** Vertical asymptote(s) at $ x = \_\_\_ $ and hole(s) corresponding to $ x = \_\_\_ $.
- **D.** There are no discontinuities.

**Instructions:**
- Select the correct option and fill in the x-values where necessary.
- Type an integer or a fraction; use commas to separate answers as needed.

**Action:**
Click the answer box to enter your selection.

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*Note: The question requires users to identify and classify the points of discontinuity for the given function, differentiating between vertical asymptotes and holes by analyzing the factored form of the denominator.*

Expert Solution

Step 1

Given:

$f (x) = \frac{x - 6}{x^{2} - 36}$

To Find:

Vertical Asymptotes and Hole

Concept:

$Vertical Asymptotes: x = c is called vertical asymptotes of f (x) if \lim_{x \to c} f (x) = \pm \infty . Hole: If f (x) has a common factor in numerator and denominator Let {(x - a)}^{m} is a factor in numerator and {(x - a)}^{n} is a factor in denominator, then (i) . if m < n, then x = a is a vertical asymptote of f (x) . (i i) . if m \geq n, t h e n x = a is a hole in the graph f (x) .$