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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 2.2.67b
Chapter 2, Problem 2.2.67b

Estimating Limits


[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.


Let f(x) = (x² - 9) / (x + 3)


b. Support your conclusions in part (a) by graphing f near c = -3 and using Zoom and Trace to estimate y-values on the graph as x → −3.

Verified step by step guidance
1
First, understand the function f(x) = (x² - 9) / (x + 3). Notice that the expression x² - 9 can be factored as (x - 3)(x + 3). This will help simplify the function.
Next, simplify the function by canceling out the common factor (x + 3) in the numerator and denominator, which gives f(x) = x - 3 for x ≠ -3. This simplification is valid except at x = -3, where the original function is undefined.
To estimate the limit as x approaches -3, use a graphing calculator to plot the function f(x) = (x² - 9) / (x + 3). Set the viewing window to focus around x = -3, such as from x = -4 to x = -2.
Utilize the Zoom feature on the graphing calculator to closely examine the behavior of the graph near x = -3. This will help you observe how the y-values change as x approaches -3 from both sides.
Finally, use the Trace feature to estimate the y-values of the graph as x approaches -3. This will provide a numerical approximation of the limit, supporting the conclusion that the limit of f(x) as x approaches -3 is the value of the simplified function at x = -3.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, we are interested in the limit of f(x) as x approaches -3, which helps us understand the behavior of the function near that point, especially since direct substitution leads to an indeterminate form.
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Graphing Functions

Graphing functions allows us to visualize their behavior and identify key features such as intercepts, asymptotes, and continuity. By using a graphing calculator, we can observe how f(x) behaves as x approaches -3, which aids in estimating the limit and understanding the function's overall shape.
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Indeterminate Forms

Indeterminate forms occur when direct substitution in a limit leads to expressions like 0/0 or ∞/∞, which do not provide clear information about the limit. In this case, substituting x = -3 into f(x) results in 0/0, prompting the need for further analysis, such as factoring or using graphical methods to resolve the limit.
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Related Practice
Textbook Question

[Technology Exercise] Roots


Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.


a. Use the Intermediate Value Theorem to show that ƒ has a zero between ―1 and 2 .

Textbook Question

Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.


lim ( 1 / x¹/³ − 1 / (x − 1)⁴/³ ) as


a. x → 0⁺

Textbook Question

Average Rates of Change


In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.


g(t)=2+cos t


b. [0,π]

Textbook Question

Limits and Continuity

On what intervals are the following functions continuous?


b. g(x) = x³/⁴

Textbook Question

Finding Limits Graphically


Which of the following statements about the function y = f(x) graphed here are true, and which are false?



b. limx→2 f(x) does not exist

Textbook Question

Average Rates of Change


In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.


g(x)=x²−2x


a. [1, 3]