Textbook Question
Determine the following limits.
lim x→0^− 2 / tan x
1. Limits and Continuity
Finding Limits Algebraically
1:34 minutesProblem 2.4.7b
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)
Verified step by step guidance1
Identify that the limit \( \lim_{x \to 1^+} f(x) \) involves approaching the point \( x = 1 \) from the right.
Recognize that a vertical asymptote at \( x = 1 \) implies that as \( x \) approaches 1 from the right, \( f(x) \) will tend towards either positive or negative infinity.
Examine the behavior of \( f(x) \) as \( x \to 1^+ \) by considering the values of \( f(x) \) for \( x \) slightly greater than 1.
Determine whether \( f(x) \) increases without bound (approaches \( +\infty \)) or decreases without bound (approaches \( -\infty \)) as \( x \to 1^+ \).
Conclude the analysis of the limit based on the direction in which \( f(x) \) tends as \( x \to 1^+ \).
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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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One-Sided Limits
One-sided limits refer to the behavior of a function as the input approaches a specific value from one side only. The notation lim x→1^+ f(x) indicates the limit of f(x) as x approaches 1 from the right (values greater than 1). Understanding one-sided limits is crucial for analyzing functions with vertical asymptotes.
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Limit Behavior Near Asymptotes
The limit behavior near vertical asymptotes is characterized by the function's tendency to increase or decrease without bound. For instance, if lim x→1^+ f(x) approaches positive infinity, it indicates that as x gets closer to 1 from the right, the function's values rise indefinitely. This behavior is essential for understanding the overall shape and characteristics of the graph.
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