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
1:19 minutesProblem 2.4.9a
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
Identify that the limit is approaching from the left side of x = -2, denoted by x → -2⁻.
Recognize that a vertical asymptote at x = -2 implies that as x approaches -2, the function h(x) tends to infinity or negative infinity.
Since the limit is from the left, consider the behavior of h(x) as x approaches -2 from values less than -2.
Examine the graph of h(x) to determine whether h(x) increases towards positive infinity or decreases towards negative infinity as x approaches -2 from the left.
Conclude the limit based on the observed behavior of h(x) as x approaches -2 from the left side.
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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→−2^− h(x) specifically refers to the limit of h(x) as x approaches -2 from the left side. 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 point from one direction only. The notation lim x→−2^− h(x) indicates a left-hand limit, meaning we are interested in the values of h(x) as x approaches -2 from values less than -2. This concept is essential for understanding how functions behave near points of discontinuity.
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