Chapter 1: Q. 59 (page 108)

Write delta-epsilon proofs for each of the limit statements in Exercises $47 - 60 .$
$\lim_{x \to 5^{+}} \sqrt{x - 5} = 0$

Short Answer

Expert verified

Delta-epsilon proof is,

Whenever $x \in (5, 5 + δ)$ , we also have $| \sqrt{x - 5} - 0 | < \in$ .

Step by step solution

Step 1. Given information

We are given,

$\lim_{x \to 5^{+}} \sqrt{x - 5} = 0$

Step 2. Writing the delta-epsilon proofs

The strategy is to write delta-epsilon proofs for the given limit statement.

Consider that $\in > 0$ , choose $δ = ϵ^{2}$ .

The limit statement $\lim_{x \to 5^{+}} \sqrt{x - 5} = 0$ means that for all $\in > 0$ , there exist $δ > 0$ such that if $x \in (5, 5 + δ)$ , then $| \sqrt{x - 5} - 0 | < ϵ .$

$5 < x < 5 + δ 5 - 5 < x - 5 < 5 + δ - 5 0 < x - 5 < δ$

This means that, when localid="1648029258219" $0 < x - 5 < δ$ , we have

$| \sqrt{x - 5} - 0 | = | \sqrt{x - 5} | = \sqrt{x - 5} < \sqrt{δ} = \sqrt{\in^{2}} = \in$

So, whenever $x \in (5, 5 + δ)$ , we also have $| \sqrt{x - 5} - 0 | < \in$ .

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Write delta-epsilon proofs for each of the limit statements in Exercises $47 - 60 .$
$\lim_{x \to 5^{+}} \sqrt{x - 5} = 0$

Short Answer

Step by step solution

Step 1. Given information

Step 2. Writing the delta-epsilon proofs

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Write delta-epsilon proofs for each of the limit statements in Exercises 47–60.limx→5+x-5=0

Short Answer

Step by step solution

Step 1. Given information

Step 2. Writing the delta-epsilon proofs

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Decision Maths

Statistics

Applied Mathematics

Geometry

Calculus

Theoretical and Mathematical Physics

Study anywhere. Anytime. Across all devices.

Write delta-epsilon proofs for each of the limit statements in Exercises $47 - 60 .$
$\lim_{x \to 5^{+}} \sqrt{x - 5} = 0$