Find the vertical, horizontal, and oblique asymptotes, if any, for the following rational function.
[tex]R(x)=\frac{5 x}{x+19}[/tex]
Find the vertical asymptotes. Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
A. The function has one vertical asymptote, [ ].
(Type an equation. Use integers or fractions for any numbers in the equation.)
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.
Answer (2)
The rational function R ( x ) = x + 19 5 x has one vertical asymptote at x = − 19 . This is determined by setting the denominator equal to zero and solving for x . Therefore, the correct choice is A: The function has one vertical asymptote, x = − 19 .
;
Find values of x where the denominator of R ( x ) is zero.
Set x + 19 = 0 .
Solve for x , obtaining x = − 19 .
The vertical asymptote is x = − 19 .
Explanation
Finding Vertical Asymptotes To find the vertical asymptotes of the rational function R ( x ) = x + 19 5 x , we need to determine the values of x for which the denominator is equal to zero, as this is where the function is undefined.
Setting Denominator to Zero We set the denominator equal to zero: x + 19 = 0
Solving for x Solving for x , we subtract 19 from both sides of the equation: x = − 19
Conclusion Therefore, the function has one vertical asymptote at x = − 19 .
Examples
Understanding asymptotes is crucial in fields like physics and engineering, where functions often model real-world phenomena. For instance, in electrical engineering, the impedance of a circuit can be represented as a rational function. Identifying vertical asymptotes helps engineers determine resonant frequencies, which are critical for designing filters and amplifiers. Similarly, in physics, understanding asymptotes can aid in modeling the behavior of particles or fields near singularities, providing insights into system stability and potential points of failure.
