Ch. 2 - Limits

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 2 - Limits

Problem 2.7.58

Chapter 2, Problem 2.7.58

Use the definitions given in Exercise 57 to prove the following infinite limits.

lim x→1^+ 1 /1 − x=−∞

Verified step by step guidance

Understand the problem: We need to prove that as x approaches 1 from the right (x → 1^+), the expression 1/(1-x) approaches negative infinity.

Consider the behavior of the denominator (1-x) as x approaches 1 from the right. Since x is slightly greater than 1, (1-x) becomes a small negative number.

Analyze the fraction 1/(1-x): As (1-x) becomes a small negative number, the fraction 1/(1-x) becomes a large negative number because dividing by a small negative number results in a large negative value.

Formalize the limit: For any large negative number M, we need to find a δ > 0 such that for all x satisfying 0 < x - 1 < δ, the inequality 1/(1-x) < M holds true.

Conclude the proof: By choosing δ = 1 - (1/M), we ensure that as x approaches 1 from the right, 1/(1-x) becomes less than any large negative number M, thus proving the limit is negative infinity.

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Key Concepts

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

Infinite Limits

Infinite limits describe the behavior of a function as it approaches a certain point, where the function's value increases or decreases without bound. In this case, as x approaches 1 from the right (1+), the function 1/(1-x) tends toward negative infinity, indicating that the values of the function decrease indefinitely.

Understanding the Function's Behavior

To analyze the limit lim x→1^+ 1/(1-x), it is essential to understand how the denominator behaves as x approaches 1 from the right. As x gets closer to 1, 1-x approaches 0, causing the fraction to grow larger in magnitude and negative, leading to the conclusion that the limit is negative infinity.

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Use the definitions given in Exercise 57 to prove the following infinite limits.lim x→1^+ 1 /1 − x=−∞

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Key Concepts

Infinite Limits

One-Sided Limits

Understanding the Function's Behavior

Use the definitions given in Exercise 57 to prove the following infinite limits.

lim x→1^+ 1 /1 − x=−∞