Evaluate limx→[infinity]f(x) and limx→−[infinity]f(x) for the rational function. Then give the horizontal asymptote of f (if any). f(x)=2x6−3x512x6−7 Select the correct choice below, and, if necessary, fill in the answer box to complete your choice. A. limx→−[infinity]2x6−3x512x6−7= (Simplify your answer.) B. The limit does not exist and is neither [infinity] or −[infinity].

Question

19-09-2023
Mathematics

High School

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Evaluate limx→[infinity]f(x) and limx→−[infinity]f(x) for the rational function. Then give the horizontal asymptote of f (if any). f(x)=2x6−3x512x6−7 Select the correct choice below, and, if necessary, fill in the answer box to complete your choice. A. limx→−[infinity]2x6−3x512x6−7= (Simplify your answer.) B. The limit does not exist and is neither [infinity] or −[infinity].

Answer :

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suparnakse26 suparnakse26 · Answer 1 · 2024-01-23T10:32:03-05:00

Final answer:

To evaluate the limits lim[x→∞]f(x) and lim[x→−∞]f(x) for the rational function f(x)=2x^6−3x^5/12x^6−7, we compare the degrees of the highest power terms in the numerator and denominator. Both limits are equal to 0. The horizontal asymptote of f(x) is y = 0.

Explanation:

To evaluate the limit limx→[infinity]f(x), we need to determine the behavior of the function as x approaches infinity. We can do this by looking at the degrees of the highest power terms in the numerator and denominator of the rational function.

In this case, the highest power term in the numerator is 2x⁶ and the highest power term in the denominator is 12x⁶. Since the degree of the numerator is less than the degree of the denominator, the limit limx→[infinity]f(x) is equal to 0.

To evaluate the limit limx→−[infinity]f(x), we can follow the same process. Again, the degrees of the highest power terms in the numerator and denominator are 2x⁶ and 12x⁶ respectively. Since the degree of the numerator is less than the degree of the denominator, the limit limx→−[infinity]f(x) is also equal to 0.

Since the limits as x approaches positive and negative infinity are both equal to 0, there is a horizontal asymptote at y = 0.

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