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.

Verified step by step guidance

1. <Step 1: Simplify the function.> Start by simplifying the given function \( f(x) = \frac{\cos x + 2\sqrt{x}}{\sqrt{x}} \). This can be rewritten as \( f(x) = \frac{\cos x}{\sqrt{x}} + 2 \).
2. <Step 2: Identify potential vertical asymptotes.> Vertical asymptotes occur where the denominator is zero and the numerator is not zero. Here, the denominator \( \sqrt{x} \) is zero when \( x = 0 \).
3. <Step 3: Analyze the limit as \( x \to 0^- \).> Since \( \sqrt{x} \) is not defined for negative \( x \), we only consider the limit from the right, \( x \to 0^+ \).
4. <Step 4: Analyze the limit as \( x \to 0^+ \).> 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.
5. <Step 5: Conclude the behavior at the asymptote.> Since \( \lim_{x \to 0^+} f(x) = \infty \), there is a vertical asymptote at \( x = 0 \). The function approaches infinity as \( x \to 0^+ \).

Key Concepts

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.