Determine limx→∞f(x) and limx→−∞f(x) for the following functions. Then give the horizontal asymptotes of f (if any).
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First, identify the highest power of x in both the numerator and the denominator. In the given function f(x) = \frac{4x^3+1}{2x^3+\sqrt{16x^6+1}}, the highest power of x in the numerator is x^3 and in the denominator is x^3 (from the term \sqrt{16x^6+1}).
To simplify the expression, divide every term in the numerator and the denominator by x^3, the highest power of x in the denominator.
After dividing, the function becomes f(x) = \frac{4 + \frac{1}{x^3}}{2 + \sqrt{16 + \frac{1}{x^6}}}.
Now, evaluate the limit as x approaches infinity. As x approaches infinity, the terms \frac{1}{x^3} and \frac{1}{x^6} approach 0. Thus, the function simplifies to \frac{4}{2 + \sqrt{16}}.
Similarly, evaluate the limit as x approaches negative infinity. The simplification process is the same, and the function again simplifies to \frac{4}{2 + \sqrt{16}}. Therefore, the horizontal asymptote is y = \frac{4}{2 + \sqrt{16}}.
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