Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.

f(x)=cos x+2√x / √x.

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Step 1: Simplify the function f(x) = \(\frac{\cos x + 2\sqrt{x}\)}{\(\sqrt{x}\)}.

Step 2: Divide each term in the numerator by \(\sqrt{x}\) to get f(x) = \(\frac{\cos x}{\sqrt{x}\)} + 2.

Step 3: Analyze \(\lim\)_{x \(\to\) \(\infty\)} f(x). As x approaches infinity, \(\frac{\cos x}{\sqrt{x}\)} approaches 0 because \(\cos\) x is bounded between -1 and 1, while \(\sqrt{x}\) grows without bound.

Step 4: Therefore, \(\lim\)_{x \(\to\) \(\infty\)} f(x) = 0 + 2 = 2, indicating a horizontal asymptote at y = 2 as x approaches infinity.

Step 5: Analyze \(\lim\)_{x \(\to\) -\(\infty\)} f(x). Since \(\sqrt{x}\) is not defined for negative x, the limit as x approaches negative infinity does not exist, and there is no horizontal asymptote in this direction.

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주요 개념

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Limits at Infinity

Limits at infinity involve evaluating the behavior of a function as the input approaches positive or negative infinity. This analysis helps determine the end behavior of the function, which is crucial for identifying horizontal asymptotes. For example, if the limit of f(x) as x approaches infinity exists and is a finite number, it indicates a horizontal asymptote at that value.

Horizontal Asymptotes

Horizontal asymptotes are lines that a graph approaches as x approaches infinity or negative infinity. They represent the value that the function approaches but may never actually reach. To find horizontal asymptotes, one typically evaluates the limits of the function at both ends of the x-axis, determining if the function stabilizes at a particular value.

Rational Functions and Simplification

Rational functions are ratios of polynomials, and their behavior at infinity can often be simplified by dividing the numerator and denominator by the highest power of x present. In the given function, simplifying helps to clearly see how the function behaves as x approaches infinity or negative infinity, which is essential for accurately determining limits and asymptotes.

Analyze lim x→∞ f(x) and lim x→−∞ f(x), and then identify any horizontal asymptotes.f(x)=cos x+2√x / √x.