Textbook Question
Determine whether the following statements are true and give an explanation or counterexample.
The line x=−1 is a vertical asymptote of the function f(x) =x^2 − 7x + 6 / x^2 − 1.
1. Limits and Continuity
Introduction to Limits
2:13 minutesProblem 2.4.65
Textbook Question
Use analytical methods and/or a graphing utility to identify the vertical asymptotes (if any) of the following functions.
f(x)=1/ √x sec x
Verified step by step guidance1
Step 1: Identify the domain of the function. The function f(x) = \( \frac{1}{\sqrt{x} \sec x} \) is defined where both \( \sqrt{x} \) and \( \sec x \) are defined and non-zero. \( \sqrt{x} \) is defined for \( x > 0 \), and \( \sec x = \frac{1}{\cos x} \) is defined where \( \cos x \neq 0 \).
Step 2: Determine where \( \sec x \) is undefined. The secant function, \( \sec x \), is undefined where \( \cos x = 0 \). This occurs at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is an integer.
Step 3: Analyze the behavior of the function near the points where \( \sec x \) is undefined. As \( x \) approaches \( \frac{\pi}{2} + n\pi \), \( \cos x \) approaches 0, causing \( \sec x \) to approach infinity or negative infinity, leading to potential vertical asymptotes.
Step 4: Consider the behavior of \( \sqrt{x} \) as \( x \to 0^+ \). As \( x \to 0^+ \), \( \sqrt{x} \to 0^+ \), which causes the function \( f(x) \) to approach infinity, indicating a potential vertical asymptote at \( x = 0 \).
Step 5: Conclude the locations of the vertical asymptotes. Based on the analysis, the vertical asymptotes occur at \( x = 0 \) and \( x = \frac{\pi}{2} + n\pi \) for integer values of \( n \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Vertical Asymptotes
Vertical asymptotes occur in a function when the output approaches infinity as the input approaches a certain value. This typically happens when the function is undefined at that point, often due to division by zero. Identifying vertical asymptotes involves finding values of x that make the denominator zero while ensuring the numerator is not also zero at those points.
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Domain of a Function
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the function f(x) = 1/√x sec x, the domain is restricted by the square root and the secant function. Understanding the domain is crucial for identifying vertical asymptotes, as any x-value that leads to an undefined function must be considered.
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Graphing Utility
A graphing utility is a tool, often software or a calculator, that allows users to visualize functions and their behaviors. By plotting the function f(x) = 1/√x sec x, one can observe where the function approaches infinity, indicating vertical asymptotes. Graphing utilities can also help in analyzing the overall shape and trends of the function, providing insights that may not be immediately apparent through algebraic methods alone.
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