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)
1. Limits and Continuity
Finding Limits Algebraically
1:20 minutesProblem 2.4.9e
Textbook Question
The graph of ℎ in the figure has vertical asymptotes at x=−2 and x=3. Analyze the following limits. <IMAGE>
lim x→3^+ h(x)
Verified step by step guidance1
Identify the behavior of the function h(x) as x approaches 3 from the right (x → 3^+).
Recognize that a vertical asymptote at x = 3 indicates that the function h(x) becomes unbounded as x approaches 3.
Since we are considering the limit from the right (x → 3^+), observe the direction in which h(x) tends as x gets closer to 3 from values greater than 3.
Determine whether h(x) approaches positive infinity or negative infinity as x approaches 3 from the right, based on the graph's behavior near the asymptote.
Conclude the limit by stating that lim x→3^+ h(x) is either positive infinity or negative infinity, depending on the observed behavior.
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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 h has vertical asymptotes at x = -2 and x = 3, indicating that as x approaches these values, the function's output becomes unbounded.
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Limits
A limit describes the behavior of a function as the input approaches a particular value. The notation lim x→3^+ h(x) specifically refers to the limit of h(x) as x approaches 3 from the right side, which is crucial for understanding how the function behaves near the vertical asymptote at x = 3.
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One-Sided Limits
One-sided limits evaluate the behavior of a function as the input approaches a specific point from one direction only. The notation lim x→3^+ h(x) indicates a right-hand limit, which helps determine the value that h(x) approaches as x gets closer to 3 from values greater than 3, providing insight into the function's behavior near the asymptote.
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