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)
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1. Limits and Continuity
Finding Limits Algebraically
1:48 minutesProblem 2.4.7c
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 the behavior of the function \( f(x) \) as \( x \) approaches 1 from the left (\( x \to 1^- \)).
Identify the behavior of the function \( f(x) \) as \( x \) approaches 1 from the right (\( x \to 1^+ \)).
Determine if the left-hand limit and the right-hand limit as \( x \to 1 \) are equal or not.
If the left-hand limit and right-hand limit are not equal, conclude that the limit \( \lim_{x \to 1} f(x) \) does not exist.
If the left-hand limit and right-hand limit are equal, state that the limit \( \lim_{x \to 1} f(x) \) is equal to that common value.
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Video duration:
1mKey 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 limit but instead diverges.
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Limits
A limit describes the behavior of a function as the input approaches a particular value. In the context of vertical asymptotes, the limit of f(x) as x approaches 1 or 2 will help determine whether f(x) approaches positive or negative infinity, which is crucial for understanding the function's behavior near these points.
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One-Sided Limits
One-sided limits are used to analyze the behavior of a function as it approaches a specific point from one side only, either the left or the right. For the limits as x approaches 1, it is important to evaluate both the left-hand limit (as x approaches 1 from values less than 1) and the right-hand limit (as x approaches 1 from values greater than 1) to fully understand the function's behavior at the asymptote.
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