Textbook Question
The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>
lim x→1^− f(x)
1. Limits and Continuity
Finding Limits Algebraically
2:06 minutesProblem 2.4.7f
Textbook Question
The graph of f in the figure has vertical asymptotes at x=1 and x=2. Analyze the following limits. <IMAGE>
lim x→2 f (x)
Verified step by step guidance1
Identify the behavior of the function \( f(x) \) as \( x \) approaches the vertical asymptote at \( x = 2 \).
Consider the limit from the left: \( \lim_{x \to 2^-} f(x) \). Analyze how \( f(x) \) behaves as \( x \) approaches 2 from values less than 2.
Consider the limit from the right: \( \lim_{x \to 2^+} f(x) \). Analyze how \( f(x) \) behaves as \( x \) approaches 2 from values greater than 2.
Determine if the left-hand limit and the right-hand limit are equal or if they diverge to \( \pm \infty \).
Conclude the overall limit \( \lim_{x \to 2} f(x) \) based on the behavior of the left-hand and right-hand limits.
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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 the graph of a function where the function approaches infinity or negative infinity as the input approaches a certain value. In this case, the function f has vertical asymptotes at x=1 and x=2, indicating that as x approaches these values, f(x) does not settle at a finite value but instead diverges.
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Limits
A limit describes the behavior of a function as the input approaches a particular point. In the context of the question, evaluating the limit as x approaches 2 involves determining the value that f(x) approaches as x gets closer to 2, which is crucial for understanding the function's behavior near its vertical asymptote.
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One-Sided Limits
One-sided limits are used to analyze the behavior of a function as it approaches a specific point from one side only, either from the left (denoted as lim x→2⁻ f(x)) or from the right (lim x→2⁺ f(x)). This distinction is important when dealing with vertical asymptotes, as the function may behave differently when approaching the asymptote from either direction.
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