Textbook Question
Infinite Limits
Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.
b. lim x→0⁻ 2 / (3x¹/³)
1. Limits and Continuity
Introduction to Limits
1:45 minutesProblem 2.2.72b
Textbook Question
Estimating Limits
[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.
Let F(x)=(x² + 3x + 2)/(2−|x|)
b. Support your conclusion in part (a) by graphing F near c = -2 and using Zoom and Trace to estimate y-values on the graph as x→−2.
Verified step by step guidance1
First, understand the function F(x) = (x² + 3x + 2)/(2−|x|). This function involves a rational expression where the denominator includes an absolute value, which can affect the behavior of the function near certain points.
Identify the point of interest, c = -2, where you need to estimate the limit of F(x) as x approaches -2. This involves analyzing the behavior of the function as x gets very close to -2 from both sides.
Use a graphing calculator to plot the function F(x). Set the viewing window to include x-values around -2, such as from -2.5 to -1.5, to observe the behavior of the function near the point of interest.
Utilize the Zoom feature on the graphing calculator to focus closely on the region around x = -2. This will help you see how the function behaves as x approaches -2 from the left and right.
Use the Trace feature on the graphing calculator to move along the graph and observe the y-values as x approaches -2. This will allow you to estimate the limit by seeing if the y-values approach a particular number as x gets closer to -2.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Limits
A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near points of interest, including points of discontinuity or where they are not defined. For example, evaluating the limit of F(x) as x approaches -2 allows us to determine the function's value or behavior at that point.
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Graphing Functions
Graphing functions involves plotting the values of a function on a coordinate plane to visualize its behavior. This technique is essential for estimating limits, as it allows one to observe how the function behaves as it approaches a specific x-value. Using tools like graphing calculators can enhance this process by providing precise visual representations and enabling features like Zoom and Trace to analyze specific points.
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Absolute Value Functions
Absolute value functions, denoted as |x|, measure the distance of a number from zero on the number line, resulting in non-negative outputs. In the context of the function F(x), the presence of the absolute value affects the function's behavior, particularly around points where x is negative. Understanding how absolute values influence the function is crucial for accurately estimating limits and interpreting the graph.
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