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Answer :
Final Answer:
1. Vertical Asymptote(s): None
2. Hole(s): x = 0
3. Horizontal Asymptote: y = 0
4. (X)-Intercept: None
Explanation:
To determine the vertical asymptotes, holes, horizontal asymptote, and x-intercepts of the rational function R(x), we need to analyze its characteristics.
Vertical Asymptotes: These occur where the denominator of the rational function becomes zero and the numerator is not zero at the same point. In this case, the denominator is a polynomial of degree 5, and the numerator is a polynomial of degree 4. To find the vertical asymptotes, we need to solve for x when the denominator equals zero, but after examination, it becomes evident that there are no real values of x that make the denominator zero while keeping the numerator nonzero. Therefore, there are no vertical asymptotes.
Holes: Holes exist where both the numerator and denominator of the rational function have a common factor that can be canceled out. In this function, if you factor out the common factor of 5x from both the numerator and denominator, you are left with a hole at x = 0.
Horizontal Asymptote: To find the horizontal asymptote, we compare the degrees of the numerator and denominator. Since the degree of the numerator is less than the degree of the denominator, the horizontal asymptote is y = 0 (the x-axis).
X-Intercepts: X-intercepts occur when the numerator of the rational function equals zero. However, upon examining the numerator, we find that there are no real solutions for x when setting it equal to zero.
In summary, the rational function R(x) has no vertical asymptotes, a hole at x = 0, a horizontal asymptote at y = 0, and no x-intercepts.
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