Textbook Question
Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a- f(x) and lim x→a+ f(x).
f(x) = (x^4−1)/(x^2−1)
1. Limits and Continuity
Finding Limits Algebraically
2:29 minutesProblem 78b
Textbook Question
Find the vertical asymptotes. For each vertical asymptote x=a, analyze lim x→a^− f(x) and lim x→a^+f(x).
f(x)=|1−x^2| / x(x+1)
Verified step by step guidance1
Step 1: Identify the points where the denominator is zero. Set the denominator equal to zero: x(x+1) = 0. Solve for x to find the potential vertical asymptotes.
Step 2: Solve the equation x(x+1) = 0. This gives the solutions x = 0 and x = -1, which are the potential vertical asymptotes.
Step 3: Analyze the behavior of the function as x approaches each potential vertical asymptote from the left and right. Start with x = 0.
Step 4: For x = 0, evaluate the limits: lim_{x \to 0^-} \frac{|1-x^2|}{x(x+1)} and lim_{x \to 0^+} \frac{|1-x^2|}{x(x+1)}. Consider the sign of the numerator and denominator as x approaches 0 from both sides.
Step 5: Repeat the analysis for x = -1. Evaluate the limits: lim_{x \to -1^-} \frac{|1-x^2|}{x(x+1)} and lim_{x \to -1^+} \frac{|1-x^2|}{x(x+1)}. Consider the sign of the numerator and denominator as x approaches -1 from both sides.
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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 function approaches infinity as the input approaches a certain value from either the left or the right. This typically happens at points where the function is undefined, 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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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, we analyze the left-hand limit (lim x→a^− f(x)) and the right-hand limit (lim x→a^+ f(x)) to determine the behavior of the function near the asymptote. These limits help us understand whether the function approaches positive or negative infinity.
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Piecewise Functions
The function f(x) = |1−x^2| / x(x+1) is a piecewise function due to the absolute value in the numerator. This means that the function behaves differently depending on the value of x. Understanding how to handle absolute values is crucial, as it can affect the limits and the overall behavior of the function, particularly when determining the vertical asymptotes.
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