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Chapter 7: Q. 59 (page 604)

Evaluate the limits in Exercises 55鈥�60. Use the theorems in this section to justify each step of your work.
$\lim_{x 鈫� 鈭�} \sqrt{k^{2} + k} - k$

Short Answer

Expert verified

$\lim_{x 鈫� 鈭�} \sqrt{k^{2} + k} - k = \frac{1}{2}$

Step by step solution

01

Step 1. Given information

$\lim_{x 鈫� 鈭�} \sqrt{k^{2} + k} - k$

02

Step 2. Calculate the limit

$\lim_{x 鈫� 鈭�} \sqrt{k^{2} + k} - k = \lim_{x 鈫� 鈭� 鈥�} (\frac{1}{\sqrt{1 + \frac{1}{x}} + 1})$

$\lim_{x 鈫� a} [\frac{f (x)}{g (x)}] = \frac{\lim_{x 鈫� a} f (x)}{\lim_{x 鈫� a} g (x)}, 鈥� \lim_{x 鈫� a} g (x) 鈮� 0$

role="math" localid="1649174927444"

\lim_{x 鈫� 鈭�} \sqrt{k^{2} + k} - k = \frac{\lim_{x 鈫� 鈭� 鈥�} (1)}{\lim_{x 鈫� 鈭� 鈥�} (\sqrt{1 + \frac{1}{x}} + 1)}

\lim_{x 鈫� 鈭�} \sqrt{k^{2} + k} - k = \frac{1}{2}

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Most popular questions from this chapter

What is meant by a p-series?

Express each of the repeating decimals in Exercises 71鈥�78 as a geometric series and as the quotient of two integers reduced to lowest terms.

$0.305130513051 . . .$

Examples: Construct examples of the thing(s) described in the following. Try to find examples that are different than any in the reading.

(a) A divergent series ${鈭�}_{k = 1}^{鈭�} a_{k}$ in which $a_{k} 鈫� 0$ .

(b) A divergent p-series.

(c) A convergent p-series.

Determine whether the series ${鈭�}_{k = 0}^{鈭�} \frac{5^{k + 1}}{{(- 6)}^{k}}$ converges or diverges. Give the sum of the convergent series.

Explain why the integral test may be used to analyze the given series and then use the test to determine whether the series converges or diverges.

${鈭�}_{k = 4}^{鈭�} 鈥� \frac{2 k 鈭� 4}{k^{2} 鈭� 4 k + 3}$

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