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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.2.72b
Chapter 2, Problem 2.2.72b

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 guidance
1
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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Related Practice
Textbook Question

Horizontal and Vertical Asymptotes


Use limits to determine the equations for all vertical asymptotes.


x² + x ― 6

c. y = ------------------

x² + 2x ― 8

Textbook Question

Limits and Continuity

On what intervals are the following functions continuous?


c. h(x) = x⁻²/³

Textbook Question

Infinite Limits


Find the limits in Exercises 37–48. Write ∞ or −∞ where appropriate.


b. lim x→0⁻ 2 / (3x¹/³)

Textbook Question

Exercises 5–10 refer to the function

f(x) = { x² − 1, −1 ≤ x < 0

2x, 0 < x < 1

1, x = 1

−2x + 4, 1 < x < 2

0, 2 < x < 3

graphed in the accompanying figure.

<IMAGE>

b. Does lim x → −1⁺ f (x) exist?

Textbook Question

Suppose limx→4 f(x) = 0 and lim x→4 g(x) = −3. Find


c. limx→4 (g(x))²

Textbook Question

Suppose limx→b f(x) = 7 and lim x→b g(x) = −3. Find


b. limx→b f(x)⋅g(x)