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
1:22 minutesProblem 2.4.9d
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
Step 1: Understand the concept of a vertical asymptote. A vertical asymptote at x = a means that as x approaches a, the function h(x) tends to infinity or negative infinity.
Step 2: Identify the behavior of the function h(x) as x approaches the vertical asymptote from the left side (x → 3⁻). This involves analyzing the graph to see if h(x) approaches positive or negative infinity.
Step 3: Recall that the limit of h(x) as x approaches 3 from the left (x → 3⁻) is determined by the behavior of h(x) near x = 3. If h(x) increases without bound, the limit is positive infinity. If it decreases without bound, the limit is negative infinity.
Step 4: Examine the graph near x = 3 from the left side to determine the direction in which h(x) is heading. This will help you conclude whether the limit is positive or negative infinity.
Step 5: Conclude the analysis by stating the limit based on the observed behavior of h(x) as x approaches 3 from the left. This involves stating whether the limit is positive infinity, negative infinity, or 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 has vertical asymptotes at x = -2 and x = 3, indicating that as x approaches these values, h(x) will diverge to infinity or negative infinity.
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Limits
A limit describes the behavior of a function as the input approaches a particular value. The notation lim x→c f(x) indicates the value that f(x) approaches as x gets closer to c. Understanding limits is crucial for analyzing the behavior of functions near points of discontinuity, such as vertical asymptotes.
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One-Sided Limits
One-sided limits evaluate the behavior of a function as the input approaches a specific value from one side only. The notation lim x→c− f(x) refers to the limit as x approaches c from the left. This concept is particularly important when analyzing functions with vertical asymptotes, as the left-hand and right-hand limits may yield different results.
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