Chapter 1: Q. 1 (page 108)

Explain why it makes intuitive sense that $lim_{x \to c} x = c$ for any real number c. Then use a delta–epsilon argument to prove it.

Short Answer

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Step by step solution

Step 1. Given information

We have to explain why it makes intuitive sense that $lim_{x \to c} x = c$ for any real number c. Then use a delta–epsilon argument to prove it.

Step 2. Explanation

Substitute c for x in the limit,

$lim_{x \to c} x = c$

For the limit statement $lim_{x \to c} f^{'} (x) = L$ , the delta-epsilon statement is

For all $\in> 0$ , there exist $δ > 0$ such that whenever $x \in (c - δ, c) \cup (c, c + δ) guarantees f (x) \in (L - \in, L + \in)$

Fof every x satisfying $0 < | x - c | < δ$ , every f(x) satisfies $| f (x) - L | < ϵ$

$f (x) = x and L = c$

Choose $δ =\in$

$\begin{matrix} | f (x) - L | = | x - c | \\ < δ \\ =\in \end{matrix}$

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Explain why it makes intuitive sense that $lim_{x \to c} x = c$ for any real number c. Then use a delta–epsilon argument to prove it.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

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Explain why it makes intuitive sense thatlimx→c x=c for any real number c. Then use a delta–epsilon argument to prove it.

Short Answer

Step by step solution

Step 1. Given information

Step 2. Explanation

One App. One Place for Learning.

Most popular questions from this chapter

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Explain why it makes intuitive sense that $lim_{x \to c} x = c$ for any real number c. Then use a delta–epsilon argument to prove it.