Textbook Question
Complete the following steps for the given functions.
c. Graph f and all of its asymptotes with a graphing utility. Then sketch a graph of the function by hand, correcting any errors appearing in the computer-generated graph.
1. Limits and Continuity
Finding Limits Algebraically
1:07 minutesProblem 2.4.7d
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)
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Identify that the limit is approaching from the left side of x = 2, denoted as x \to 2^-.
Recognize that the function f(x) has a vertical asymptote at x = 2, which means the function's values increase or decrease without bound as x approaches 2.
Since the limit is from the left, consider the behavior of f(x) as x approaches 2 from values less than 2.
Analyze the graph of f(x) to determine whether the function approaches positive infinity or negative infinity as x approaches 2 from the left.
Conclude that the limit \( \lim_{x \to 2^-} f(x) \) is either \( +\infty \) or \( -\infty \) based on the observed behavior of the graph.
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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. The notation lim x→2^− f(x) specifically refers to the limit of f(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 left-hand limit, denoted as lim x→2^− f(x), examines the values of f(x) as x approaches 2 from values less than 2. This concept is essential for understanding how functions behave near vertical asymptotes.
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