Ch. 2 - Limits and Continuity

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 2 - Limits and Continuity

Problem 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) = ∞

Verified step by step guidance

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.

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.

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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Textbook Question

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

b. limx→c 2f(x)g(x)

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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.

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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))

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Textbook Question

Limits of quotients

Find the limits in Exercises 23–42.

limx→1 (x −1) / (√(x + 3) − 2)

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Textbook Question

Limits and Infinity

Find the limits in Exercises 37–46.