Textbook Question
Determine the following limits.
c. lim x→4 x − 5 / (x − 4)^2
1. Limits and Continuity
Finding Limits Algebraically
2:38 minutesProblem 2.4.9c
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→−2 h(x)
Verified step by step guidance1
Step 1: Identify the type of limit problem. Since the problem involves a vertical asymptote at x = -2, we are dealing with a limit where the function approaches infinity or negative infinity.
Step 2: Understand the behavior of the function near the asymptote. As x approaches -2, the function h(x) will either increase or decrease without bound.
Step 3: Consider the direction of approach. Determine if you need to evaluate the limit from the left (x approaches -2 from the left, denoted as x → -2⁻) or from the right (x approaches -2 from the right, denoted as x → -2⁺).
Step 4: Analyze the graph of h(x) near x = -2. Observe whether the function values are increasing towards positive infinity or decreasing towards negative infinity as x approaches -2 from either side.
Step 5: Conclude the limit based on the behavior observed. If the function approaches positive infinity from both sides, the limit is positive infinity. If it approaches negative infinity, the limit is negative infinity. If the behavior differs from each side, the limit does not exist.
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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(x) has vertical asymptotes at x = -2 and x = 3, indicating that as x approaches these values, h(x) will either increase or decrease without bound.
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Limits
A limit describes the behavior of a function as the input approaches a particular value. In the context of the question, evaluating the limit of h(x) as x approaches -2 involves determining what value h(x) approaches as x gets closer to -2, which is critical for understanding the function's behavior near its vertical asymptote.
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One-Sided Limits
One-sided limits refer to the limits of a function as the input approaches a specific value from one side only, either the left or the right. For the limit lim x→−2 h(x), it is important to consider both the left-hand limit (as x approaches -2 from values less than -2) and the right-hand limit (as x approaches -2 from values greater than -2) to fully understand the behavior of h(x) near the asymptote.
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