Give the equations of any​ vertical, horizontal, or oblique asymptotes for the graph of the rational function.
f left parenthesis x right parenthesis equals StartFraction 5 minus 5 x Over 5 x plus 3 EndFraction
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Part 1
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The equation of the vertical asymptote is xequals
  
enter your response here.
​(Type an integer or a fraction. Simplify your​ answer.)
B.
There is no vertical asymptote.
Part 2
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The equation of the horizontal asymptote is yequals
  
enter your response here.
​(Type an integer or a fraction. Simplify your​ answer.)
B.
There is no horizontal asymptote.
Part 3
Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your choice.
A.
The equation of the oblique asymptote is yequals
  
enter your response here.
​(Type an integer or a fraction. Simplify your​ answer.)
B.
There is no oblique asymptote.

To find the asymptotes for the rational function \( f(x) = \frac{5 - 5x}{5x + 3} \), we will examine vertical, horizontal, and oblique asymptotes.

### Part 1: Vertical Asymptote

Vertical asymptotes occur where the denominator is zero (and the numerator is not zero at the same point).

1. Set the denominator equal to zero:
   \[
   5x + 3 = 0
   \]
   Solving for \( x \):
   \[
   5x = -3 \implies x = -\frac{3}{5}
   \]

**Answer**: A. The equation of the vertical asymptote is \( x = -\frac{3}{5} \).

### Part 2: Horizontal Asymptote

To find horizontal asymptotes, we compare the degrees of the polynomial in the numerator and denominator.

- The degree of the numerator (5 - 5x) is 1 (linear).
- The degree of the denominator (5x + 3) is also 1 (linear).

When the degrees are the same, the horizontal asymptote is given by the ratio of the leading coefficients:
\[
\text{Horizontal Asymptote: } y = \frac{-5}{5} = -1
\]

**Answer**: A. The equation of the horizontal asymptote is \( y = -1 \).

### Part 3: Oblique Asymptote

Oblique (or slant) asymptotes occur when the degree of the numerator is exactly one greater than the degree of the denominator. Since both degrees are equal here, there is no oblique asymptote.

**Answer**: B. There is no oblique asymptote.

### Summary of Answers:
1. Vertical Asymptote: \( x = -\frac{3}{5} \)
2. Horizontal Asymptote: \( y = -1 \)
3. Oblique Asymptote: None

Do you want any more details or have any questions? Here are some related questions you might find interesting:

1. How do you determine the degree of a polynomial?
2. What is the significance of asymptotes in graphing rational functions?
3. How can you sketch the graph of the function given its asymptotes?
4. What happens to the function as \( x \) approaches the vertical asymptote?
5. Can a rational function have more than one vertical asymptote? 

**Tip**: Always check both the numerator and denominator to ensure there are no common factors that could affect the asymptotes.

Give the equations of any vertical, horizontal, or oblique asymptotes for the graph of the rational function f(x) = (5 - 5x) / (5x + 3).

Explore the asymptotes of the rational function f(x) = (5 - 5x) / (5x + 3). This includes finding the vertical asymptote x = -3/5, the horizontal asymptote y = -1, and noting the absence of an oblique asymptote. Ideal for high school students learning about rational functions.

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