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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 2.6.96
Chapter 2, Problem 2.6.96

Use formal definitions to prove the limit statements in Exercises 93–96.


lim x → −5 (1 / (x + 5)²) = ∞

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To prove that \( \lim_{x \to -5} \frac{1}{(x + 5)^2} = \infty \), we need to use the formal definition of a limit approaching infinity. This means for every positive number \( M \), there exists a \( \delta > 0 \) such that if \( 0 < |x + 5| < \delta \), then \( \frac{1}{(x + 5)^2} > M \).
Start by considering the expression \( \frac{1}{(x + 5)^2} > M \). This inequality can be rewritten as \( (x + 5)^2 < \frac{1}{M} \).
To find \( \delta \), solve the inequality \( |x + 5| < \sqrt{\frac{1}{M}} \). This gives us the condition for \( \delta \) in terms of \( M \).
Set \( \delta = \sqrt{\frac{1}{M}} \). This choice of \( \delta \) ensures that whenever \( 0 < |x + 5| < \delta \), the inequality \( \frac{1}{(x + 5)^2} > M \) holds true.
Conclude that for every \( M > 0 \), there exists a \( \delta = \sqrt{\frac{1}{M}} \) such that if \( 0 < |x + 5| < \delta \), then \( \frac{1}{(x + 5)^2} > M \). Therefore, \( \lim_{x \to -5} \frac{1}{(x + 5)^2} = \infty \) is proven using the formal definition.

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Key Concepts

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

Limit Definition

The formal definition of a limit states that for a function f(x), the limit as x approaches a value a is L if, for every ε > 0, there exists a δ > 0 such that whenever 0 < |x - a| < δ, it follows that |f(x) - L| < ε. This definition is crucial for proving limit statements rigorously.
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Definition of the Definite Integral

Behavior of Functions Near Asymptotes

Understanding how functions behave near vertical asymptotes is essential for limit proofs. In this case, as x approaches -5, the denominator (x + 5) approaches zero, causing the function (1 / (x + 5)²) to increase without bound, leading to the conclusion that the limit is infinity.
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Graphs of Exponential Functions

Epsilon-Delta Proofs

Epsilon-delta proofs are a method used to rigorously establish limits. In this context, one must show that for any large number M (representing infinity), there exists a corresponding δ such that if x is within δ of -5 (but not equal to -5), then (1 / (x + 5)²) exceeds M, confirming the limit statement.
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Related Practice
Textbook Question

In Exercises 77–80, find a function that satisfies the given conditions and sketch its graph. (The answers here are not unique. Any function that satisfies the conditions is acceptable. Feel free to use formulas defined in pieces if that will help.)


lim x → ±∞ k(x) = 1, lim x → 1⁻ k(x) = ∞, and lim x → 1⁺ k(x) = −∞

Textbook Question

Suppose limx→c f(x) = 5 and lim x→c g(x) = −2. Find


b. limx→c 2f(x)g(x)

Textbook Question

Using the Sandwich Theorem


a. Suppose that the inequalities 1/2 − x² / 24 < (1 − cos x)/ x² < 1/2 hold for values of x close to zero, except for x = 0 itself. (They do, as you will see in Section 9.9.) What, if anything, does this tell you about limx→0 (1 −cos x)/ x²?


Give reasons for your answer.


[Technology Exercise] b. Graph the equations y=(1/2) − (x²/24), y = (1 - cos x) / x², and y = 1/2 together for −2 ≤ x ≤2. Comment on the behavior of the graphs as x→0.

Textbook Question

Finding Limits


In Exercises 3–8, find the limit of each function (a) as x → ∞ and (b) as x → −∞. (You may wish to visualize your answer with a graphing calculator or computer.)


g(x) = 1/(2 + (1/x))

Textbook Question

Limits of quotients


Find the limits in Exercises 23–42.


limx→1 (x −1) / (√(x + 3) − 2)

Textbook Question

Limits and Infinity


Find the limits in Exercises 37–46.


sin x

lim ------------- ( If you have a grapher, try graphing

x→∞ |x| the function for ―5 ≤ x ≤ 5 ) .