Chapter 1: Q. 69 (page 108)

Prove each of the limit statements in Exercises67–72. You
will have to bound $δ$ .
$\lim_{x \to 5} (x^{2} - 6 x + 7) = 2$

Short Answer

Expert verified

The given limit $\lim_{x \to 5} (x^{2} - 6 x + 7) = 2$ is proved.

Step by step solution

Step 1. Given information.

We are given,

$\lim_{x \to 5} (x^{2} - 6 x + 7) = 2$

Step 2. Proving the limit

Given $ε > 0$ , choose $δ = \min (1, \frac{ε}{5})$ .

Then if $0 < | x - 5 | < δ$ , we have,

$|(x^{2} - 6 x + 7) - 2| = |x^{2} - 6 x + 5| = | x (x - 5) - 1 (x - 5) | = | (x - 5) (x - 1) | = | x - 5 | | x - 1 | < δ | x - 1 |$

Step 3. Proving the limit

Now from, $0 < | x - 5 | < δ$ ,

$x - 5 < 1 x < 1 + 5 x < 6$

Therefore,

$< δ | 6 - 1 | \leq δ (5) \leq (\frac{ε}{5}) 5 \leq ε$

Hence Proved.

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Prove each of the limit statements in Exercises67–72. You
will have to bound $δ$ .
$\lim_{x \to 5} (x^{2} - 6 x + 7) = 2$

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Proving the limit

Step 3. Proving the limit

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Prove each of the limit statements in Exercises67–72. Youwill have to bound δ.limx→5x2-6x+7=2

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Proving the limit

Step 3. Proving the limit

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Prove each of the limit statements in Exercises67–72. You
will have to bound $δ$ .
$\lim_{x \to 5} (x^{2} - 6 x + 7) = 2$