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
1:23 minutesProblem 2.4.9b
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
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 direction of approach. The limit x → -2^+ indicates that we are approaching x = -2 from the right side (values greater than -2).
Step 3: Analyze the behavior of h(x) as x approaches -2 from the right. Since there is a vertical asymptote at x = -2, observe whether h(x) increases towards positive infinity or decreases towards negative infinity.
Step 4: Use the graph to determine the behavior. Look at the graph of h(x) near x = -2 from the right side to see if the function is going upwards or downwards.
Step 5: Conclude the limit based on the observed behavior. If h(x) goes to positive infinity, the limit is positive infinity. If it goes to negative infinity, the limit is negative infinity.
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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, the function's output becomes unbounded.
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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→−2^+ h(x) indicates that we are interested in the limit of h(x) as x approaches -2 from the right (values greater than -2), which helps in understanding the function's behavior near the asymptote.
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Limit Behavior Near Asymptotes
The behavior of limits near vertical asymptotes is crucial for understanding how functions behave at points where they are undefined. As x approaches a vertical asymptote, the function typically tends to either positive or negative infinity, which can be determined by analyzing the function's values just before and after the asymptote.
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