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Answer :
Final answer:
The function f(x) = 41x^6 - 2x^2 does not have horizontal asymptotes because as x approaches infinity, the highest degree term, which is x^6, causes the function's value to go to infinity.
Explanation:
The function f(x) = 41x6 - 2x2 is a polynomial function. By their nature, polynomial functions of degree one or higher do not have horizontal asymptotes since as x approaches ±∞, the value of the polynomial will also increase or decrease without bound.
For the given function, the term with the highest degree is x6, which dominates the behavior of the function as x approaches infinity, causing the function's value to also approach infinity. Therefore, the correct answer is that there are no horizontal asymptotes for this function.
When considering the horizontal asymptotes of a rational function, we look at the degrees of the polynomials in the numerator and denominator. For example, in a function like f(x) = (x2 + 3) / (x2 + 4), since the degrees of the numerator and the denominator are equal, the horizontal asymptote can be found by dividing the leading coefficients, which gives y = 1. However, this rule does not apply to non-rational functions such as the one we are examining.
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