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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 9
Chapter 2, Problem 9

Finding Deltas Graphically


In Exercises 7–14, use the graphs to find a δ>0 such that |f(x)−L| <ε whenever 0< |x−c| <δ.


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1
Understand the problem: We need to find a δ > 0 such that |f(x) - L| < ε whenever 0 < |x - c| < δ. This is related to the definition of the limit of a function as x approaches c.
Identify the limit L and the point c from the graph. The limit L is the value that f(x) approaches as x gets closer to c.
Choose an ε > 0, which represents how close f(x) should be to L. This is typically given or can be chosen based on the graph's behavior.
Examine the graph to determine the interval around c where |f(x) - L| < ε. This involves finding the range of x values where the function stays within ε of L.
Determine δ by finding the maximum distance from c within the interval identified in the previous step. δ is the largest value such that 0 < |x - c| < δ ensures |f(x) - L| < ε.

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Key Concepts

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

Limits

In calculus, a limit describes the behavior of a function as its input approaches a certain value. It is fundamental for understanding continuity and the concept of convergence. The notation 'lim x→c f(x) = L' indicates that as x gets closer to c, f(x) approaches L. This concept is crucial for defining derivatives and integrals.
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Epsilon-Delta Definition of a Limit

The epsilon-delta definition formalizes the concept of limits in calculus. It states that for every ε (epsilon) > 0, there exists a δ (delta) > 0 such that if 0 < |x - c| < δ, then |f(x) - L| < ε. This definition provides a rigorous way to prove that a function approaches a limit, ensuring precision in mathematical analysis.
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Graphical Interpretation of Limits

Graphically, limits can be understood by observing the behavior of a function's graph near a specific point. The limit L is the value that the function approaches as x approaches c. By analyzing the graph, one can visually determine the appropriate δ for a given ε, helping to illustrate the relationship between the function's values and its limit.
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Related Practice
Textbook Question

Limits and Continuity


Suppose the functions ƒ(x) and g(x) are defined for all x and that lim (x → 0) ƒ(x) = 1/2 and lim (x → 0) g(x) = √2. Find the limits as x → 0 of the following functions.


f. [ƒ(x) • cos x ] / x―1

255
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Textbook Question

Finding Limits

In Exercises 9–24, find the limit or explain why it does not exist.



lim x→a (x² ― a²)/(x⁴ ― a⁴)

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Textbook Question

Limits and Continuity

In Exercises 5 and 6, find the value that lim (x→0) g(x) must have if the given limit statements hold.


lim (x lim g(x)) = 2

x→-4 x→0

188
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Textbook Question

The accompanying figure shows the plot of distance fallen versus time for an object that fell from the lunar landing module a distance 80 m to the surface of the moon.

         

a. Estimate the slopes of the secant lines PQ₁, PQ₂, PQ₃, and PQ₄, arranging them in a table like the one in Figure 2.6.


b. About how fast was the object going when it hit the surface?


290
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Textbook Question

Limits and Continuity


In Exercises 5 and 6, find the value that lim (x→0) g(x) must have if the given limit statements hold.


lim ((4―g(x)) / x ) = 1

x→0

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Textbook Question

Finding Limits

In Exercises 9–24, find the limit or explain why it does not exist.



lim h →0 ((x + h)² ― x²)/h

303
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