Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).

f(x)=cos x+2√x / √x.

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Start by simplifying the given function \( f(x) = \frac{\cos x + 2\sqrt{x}}{\sqrt{x}} \). This can be rewritten as \( f(x) = \frac{\cos x}{\sqrt{x}} + 2 \).

Vertical asymptotes occur where the denominator is zero and the numerator is not zero. Here, the denominator \( \sqrt{x} \) is zero when \( x = 0 \).

Since \( \sqrt{x} \) is not defined for negative \( x \), we only consider the limit from the right, \( x \to 0^+ \).

Evaluate \( \lim_{x \to 0^+} \frac{\cos x}{\sqrt{x}} + 2 \). As \( x \to 0^+ \), \( \sqrt{x} \to 0^+ \) and \( \cos x \to 1 \), so the term \( \frac{\cos x}{\sqrt{x}} \) approaches infinity.

Since \( \lim_{x \to 0^+} f(x) = \infty \), there is a vertical asymptote at \( x = 0 \). The function approaches infinity as \( x \to 0^+ \).

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주요 개념

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Vertical Asymptotes

Vertical asymptotes occur in a function when the function approaches infinity or negative infinity as the input approaches a certain value from either the left or the right. This typically happens when the denominator of a rational function approaches zero while the numerator remains non-zero. Identifying vertical asymptotes is crucial for understanding the behavior of the function near those points.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In the context of vertical asymptotes, evaluating the left-hand limit (lim x→a^− f(x)) and the right-hand limit (lim x→a^+ f(x)) helps determine the behavior of the function as it nears the asymptote. This analysis is essential for understanding how the function behaves around points of discontinuity.

Rational Functions

A rational function is a ratio of two polynomials, which can exhibit vertical asymptotes when the denominator equals zero. In the given function f(x) = (cos x + 2√x) / √x, the denominator √x must be analyzed to find points where it becomes zero, leading to potential vertical asymptotes. Understanding the structure of rational functions is key to identifying their asymptotic behavior.

Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).f(x)=cos x+2√x / √x.