Chapter 7: Q. 60 (page 604)

Evaluate the limits in Exercises 55–60. Use the theorems in this section to justify each step of your work.
$\lim_{k \to \infty} (\sqrt{k^{3} + k + 1} - \sqrt{k^{3} - k - 1})$

Short Answer

Expert verified

$\lim_{k \to \infty} (\sqrt{k^{3} + k + 1} - \sqrt{k^{3} - k - 1}) = 0$

Step by step solution

Step 1. Given information

$\lim_{k \to \infty} (\sqrt{k^{3} + k + 1} - \sqrt{k^{3} - k - 1})$

Step 2. Calculate integrate

$\lim_{k \to \infty} (\sqrt{k^{3} + k + 1} - \sqrt{k^{3} - k - 1}) = \lim_{k \to \infty} (\sqrt{k^{3} + k + 1} - \sqrt{k^{3} - k - 1} \times \frac{(\sqrt{k^{3} + k + 1} + \sqrt{k^{3} - k - 1})}{(\sqrt{k^{3} + k + 1} + \sqrt{k^{3} - k - 1})})$

$\lim_{k \to \infty} (\sqrt{k^{3} + k + 1} - \sqrt{k^{3} - k - 1}) = \frac{({(\sqrt{k^{3} + k + 1})}^{2} - {(\sqrt{k^{3} - k - 1})}^{2})}{(\sqrt{k^{3} + k + 1} + \sqrt{k^{3} - k - 1})}$

$\lim_{k \to \infty} (\sqrt{k^{3} + k + 1} - \sqrt{k^{3} - k - 1}) = \frac{k^{3} + k + 1 - (k^{3} - k - 1)}{\sqrt{k^{2} + k + 1} + \sqrt{k^{3} - k - 1}}$

$\lim_{k \to \infty} (\sqrt{k^{3} + k + 1} - \sqrt{k^{3} - k - 1}) = \lim_{k \to \infty} \frac{2 k + 2}{\sqrt{k^{2} + k + 1} + \sqrt{k^{3} - k - 1}}$

$\lim_{k \to \infty} (\sqrt{k^{3} + k + 1} - \sqrt{k^{3} - k - 1}) = 0$

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Evaluate the limits in Exercises 55–60. Use the theorems in this section to justify each step of your work.
$\lim_{k \to \infty} (\sqrt{k^{3} + k + 1} - \sqrt{k^{3} - k - 1})$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculate integrate

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Evaluate the limits in Exercises 55–60. Use the theorems in this section to justify each step of your work. limk→∞k3+k+1-k3-k-1

Short Answer

Step by step solution

Step 1. Given information

Step 2. Calculate integrate

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

Mechanics Maths

Decision Maths

Pure Maths

Applied Mathematics

Statistics

Study anywhere. Anytime. Across all devices.

Evaluate the limits in Exercises 55–60. Use the theorems in this section to justify each step of your work.
$\lim_{k \to \infty} (\sqrt{k^{3} + k + 1} - \sqrt{k^{3} - k - 1})$