Q. 2 Explain why it makes intuitive s... [FREE SOLUTION]

Chapter 1: Q. 2 (page 108)

Explain why it makes intuitive sense that $lim_{x 鈫� c} 鈥� x^{2} = c^{2}$ for any real number c. Then use a delta鈥揺psilon argument to prove it. (Hint: You will need to assume that 未鈮� 1 )

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Step 1. Given information

We have to explain why it makes intuitive sense that $lim_{x 鈫� c} 鈥� x^{2} = c^{2}$ for any real number c. Then use a delta鈥揺psilon argument to prove it. (Hint: You will need to assume that 未鈮� 1 )

Step 2. Explanation

Substitute c for x in the limit,

$lim_{x 鈫� c} 鈥� x^{2} = c^{2}$

For the limit statement $lim_{x 鈫� c} 鈥� f^{鈥�} (x) = L$ , the delta-epsilon statement is

For all $鈭�&驳迟; 0$ , there exist $未 > 0$ such that whenever

$x 鈭� (c 鈭� 未, c) 鈭� (c, c + 未) guarantees f (x) 鈭� (L 鈭� 鈭�, L + 鈭�)$

For every x satisfying $0 < | x 鈭� c | < 未$ , every f(x) satisfies $| f (x) 鈭� L | < 系$

$f (x) = x^{2} and L = c^{2}$

Choose $未 = \frac{系}{2 c + 1}$

$\begin{matrix} | f (x) 鈭� L | = |x^{2} 鈭� c^{2}| \\ = | x + c | | x 鈭� c | \\ < 未 | x + c | \end{matrix}$

Now, for $0 < | x 鈭� c | < 未 and 未 < 1 substitute x = 1 + c$

$\begin{matrix} | f (x) 鈭� L | < 未 | x + c | \\ = 未 | 1 + c 2 | \\ = \frac{系}{| 1 + 2 c |} | 1 + 2 c | \\ &濒迟;鈭� \end{matrix}$

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Explain why it makes intuitive sense that $lim_{x 鈫� c} 鈥� x^{2} = c^{2}$ for any real number c. Then use a delta鈥揺psilon argument to prove it. (Hint: You will need to assume that 未鈮� 1 )

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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91影视

Explain why it makes intuitive sense that limx鈫�c鈥�x2=c2for any real number c. Then use a delta鈥揺psilon argument to prove it. (Hint: You will need to assume that 未 鈮� 1 )

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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Most popular questions from this chapter

Recommended explanations on Math Textbooks

Logic and Functions

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Study anywhere. Anytime. Across all devices.

Explain why it makes intuitive sense that $lim_{x 鈫� c} 鈥� x^{2} = c^{2}$ for any real number c. Then use a delta鈥揺psilon argument to prove it. (Hint: You will need to assume that 未鈮� 1 )