Find the vertical, horizontal, and oblique asymptotes, if any, for the following rational function:
[tex]T(x)=\frac{2 x^2}{x^4-1}[/tex]
Answer (2)
The vertical asymptotes of the function T ( x ) = x 4 − 1 2 x 2 are at x = 1 and x = − 1 , while the horizontal asymptote is at y = 0 . There are no oblique asymptotes since the degree of the numerator is less than the degree of the denominator. In total, we have x = ± 1 for verticals and y = 0 for horizontal asymptotes.
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Find vertical asymptotes by setting the denominator equal to zero: x 4 − 1 = 0 , which gives x = 1 and x = − 1 .
Determine the horizontal asymptote by evaluating the limit as x approaches infinity: lim x → ∞ x 4 − 1 2 x 2 = 0 , so y = 0 .
Check for oblique asymptotes: since the degree of the numerator (2) is less than the degree of the denominator (4), there are none.
The vertical asymptotes are x = ± 1 , the horizontal asymptote is y = 0 , and there are no oblique asymptotes: x = ± 1 , y = 0 .
Explanation
Problem Analysis We are given the rational function T ( x ) = x 4 − 1 2 x 2 and asked to find its vertical, horizontal, and oblique asymptotes. Let's start by analyzing the function to determine these asymptotes.
Finding Vertical Asymptotes Vertical asymptotes occur where the denominator of the rational function is equal to zero and the numerator is non-zero. So, we need to solve the equation x 4 − 1 = 0 .
Solving for Vertical Asymptotes We can factor the denominator as follows: x 4 − 1 = ( x 2 − 1 ) ( x 2 + 1 ) = ( x − 1 ) ( x + 1 ) ( x 2 + 1 ) Setting this equal to zero, we have ( x − 1 ) ( x + 1 ) ( x 2 + 1 ) = 0 . The real solutions are x = 1 and x = − 1 . The factor x 2 + 1 has no real roots.
Verifying Vertical Asymptotes Now we need to check if the numerator is zero at x = 1 and x = − 1 . The numerator is 2 x 2 . When x = 1 , the numerator is 2 ( 1 ) 2 = 2 = 0 . When x = − 1 , the numerator is 2 ( − 1 ) 2 = 2 = 0 . Therefore, the vertical asymptotes are x = 1 and x = − 1 .
Finding Horizontal Asymptotes To find the horizontal asymptote, we need to examine the limit of T ( x ) as x approaches infinity and negative infinity. x → ∞ lim x 4 − 1 2 x 2 x → − ∞ lim x 4 − 1 2 x 2 Since the degree of the denominator (4) is greater than the degree of the numerator (2), the limit as x approaches infinity or negative infinity is 0. Thus, the horizontal asymptote is y = 0 .
Checking for Oblique Asymptotes To find oblique asymptotes, we need to check if the degree of the numerator is exactly one greater than the degree of the denominator. In this case, the degree of the numerator is 2 and the degree of the denominator is 4. Since the degree of the numerator is not one greater than the degree of the denominator, there are no oblique asymptotes.
Final Answer In summary, the vertical asymptotes are x = 1 and x = − 1 , the horizontal asymptote is y = 0 , and there are no oblique asymptotes.
Examples
Understanding asymptotes is crucial in various fields. For instance, in physics, when modeling the behavior of a system approaching a certain limit (like the speed of light), asymptotes help define boundaries that the system can approach but never reach. Similarly, in economics, asymptotes can represent maximum production levels or minimum costs that a company can strive for but never surpass due to constraints. In computer science, they can describe the theoretical limits of algorithm performance as the input size grows infinitely large. The function T ( x ) = x 4 − 1 2 x 2 can model a scenario where the output approaches zero as the input becomes very large, indicating a diminishing return or an upper limit to a process.
