Chapter 14: Q. 13 (page 907)

Determine where the rational function
$R (x) = \frac{x^{3} + 6 x^{2} - 4 x - 24}{x^{2} + 5 x - 14}$
is undefined. Determine whether an asymptote or a hole appears at such number

Short Answer

Expert verified

The function is undefined at $c = - 7$ and has a vertical asymptote at this point and a hole at $x = - 2$

Step by step solution

Step 1. Values at which function is undefined

We can write the given function as ,

$R (x) = \frac{x^{3} + 6 x^{2} - 4 x - 24}{x^{2} + 5 x - 14} = \frac{(x + 6) (x + 2) (x - 2)}{(x - 2) (x + 7)} x - 2 = 0 \Rightarrow x = 2 x + 7 = 0 \Rightarrow x = - 7$

Step 2. Solve the limit atc=2

Now, apply the limits,

$\lim_{x \to 2} \frac{x^{3} + 6 x^{2} - 4 x - 24}{x^{2} + 5 x - 14} = \lim_{x \to 2} \frac{(x + 6) (x + 2) (x - 2)}{(x - 2) (x + 7)} = \lim_{x \to 2} \frac{x^{2} + 8 x + 12}{x + 7} = \frac{\lim_{x \to 2} (x^{2} + 8 x + 12)}{\lim_{x \to 2} (x + 7)} = \frac{\lim_{x \to 2} x^{2} + \lim_{x \to 2} 8 x + \lim_{x \to 2} 12}{\lim_{x \to 2} x + \lim_{x \to 2} 7} = \frac{2^{2} + 8 (2) + 12}{2 + 7} = - \frac{32}{8} = - 4$

Therefore, at the point $(2, - 4)$ , there will be a hole.

Step 3. Solve the limit at c=-7

On applying the limits,

$\lim_{x \to - 7} \frac{x^{3} + 6 x^{2} - 4 x - 24}{x^{2} + 5 x - 14} = \lim_{x \to - 7} \frac{(x + 6) (x + 2) (x - 2)}{(x - 2) (x + 7)} = \lim_{x \to - 7} \frac{x^{2} + 8 x + 12}{x + 7} = \frac{\lim_{x \to - 7} (x^{2} + 8 x + 12)}{\lim_{x \to - 7} (x + 7)} = \frac{\lim_{x \to - 7} x^{2} + \lim_{x \to - 7} 8 x + \lim_{x \to - 7} 12}{\lim_{x \to - 7} x + \lim_{x \to - 7} 7} = \frac{(- 7)^{2} + 8 (- 7) + 12}{(- 7) + 7} = \frac{5}{0}$

Since the limit comes out to be infinity, therefore there will be an asymptote at this value.

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Determine where the rational function
$R (x) = \frac{x^{3} + 6 x^{2} - 4 x - 24}{x^{2} + 5 x - 14}$
is undefined. Determine whether an asymptote or a hole appears at such number

Short Answer

Step by step solution

Step 1. Values at which function is undefined

Step 2. Solve the limit atc=2

Step 3. Solve the limit at c=-7

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Determine where the rational functionR(x)=x3+6x2-4x-24x2+5x-14is undefined. Determine whether an asymptote or a hole appears at such number

Short Answer

Step by step solution

Step 1. Values at which function is undefined

Step 2. Solve the limit atc=2

Step 3. Solve the limit at c=-7

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Geometry

Applied Mathematics

Probability and Statistics

Logic and Functions

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Determine where the rational function
$R (x) = \frac{x^{3} + 6 x^{2} - 4 x - 24}{x^{2} + 5 x - 14}$
is undefined. Determine whether an asymptote or a hole appears at such number