Find the vertical, horizontal, and oblique asymptotes, if any, for the following rational function.
[tex]R(x)=\frac{5 x}{x+19}[/tex]
A. The function has one vertical asymptote, [tex]x=-19[/tex].
B. The function has two vertical asymptotes. The leftmost asymptote is and the rightmost asymptote is (Type equations. Use integers or fractions for any numbers in the equations.)
C. The function has no vertical asymptote.
Find the horizontal asymptotes. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A. The function has one horizontal asymptote, [tex]\square[/tex] .
B. The function has two horizontal asymptotes. The top asymptote is [tex]\square[/tex] and the bottom asymptote is [tex]\square[/tex] .
C. The function has no horizontal asymptote.
Answer (2)
The function R ( x ) = x + 19 5 x has one vertical asymptote at x = − 19 and one horizontal asymptote at y = 5 . There are no oblique asymptotes. Thus, the correct choices are: vertical asymptote (A) and horizontal asymptote (A).
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The vertical asymptote is x = − 19 and the horizontal asymptote is y = 5 . x = − 19 , y = 5
Explanation
Problem Analysis We are given the rational function R ( x ) = x + 19 5 x and asked to find its vertical, horizontal, and oblique asymptotes.
Finding Vertical Asymptotes Vertical asymptotes occur where the denominator of the rational function is equal to zero. So, we set the denominator x + 19 equal to zero and solve for x :
x + 19 = 0 x = − 19 Thus, there is one vertical asymptote at x = − 19 .
Finding Horizontal Asymptotes Horizontal asymptotes are found by examining the limit of the function as x approaches infinity. We have: x → ∞ lim x + 19 5 x To evaluate this limit, we divide both the numerator and the denominator by x :
x → ∞ lim ( x + 19 ) / x 5 x / x = x → ∞ lim 1 + 19/ x 5 As x approaches infinity, the term 19/ x approaches 0, so the limit becomes: 1 + 0 5 = 5 Thus, there is a horizontal asymptote at y = 5 .
Checking for Oblique Asymptotes Oblique asymptotes occur when the degree of the numerator is one greater than the degree of the denominator. In this case, the degree of the numerator is 1 and the degree of the denominator is 1. Since the degrees are equal, there are no oblique asymptotes.
Final Answer Therefore, the function R ( x ) = x + 19 5 x has one vertical asymptote at x = − 19 and one horizontal asymptote at y = 5 . There are no oblique asymptotes.
Examples
Rational functions and their asymptotes are useful in modeling various real-world phenomena. For example, in pharmacology, the concentration of a drug in the bloodstream over time can be modeled by a rational function. The horizontal asymptote represents the steady-state concentration of the drug, while the vertical asymptote can indicate a toxic level. Understanding these asymptotes helps in determining safe and effective dosages.
