Chapter 1: Q. 16 (page 97)

It is false that $\lim_{x \to 1} (x + 1.01) = 2$ . Express this fact in a mathematical sentence involving $δ$ and $ε$ , to show how the formal definition of limit fails in this case.

Short Answer

Expert verified

The $δ < 0$ , the expression is false.

Step by step solution

Step 1. Given information.

The given function is $\lim_{x \to 1} (x + 1.01) = 2$ .

Step 2. Calculation.

From the given expression, we have, $c = 1, L = 2$ .

The limit expression can be written as a formal statement as below,

For all epsilon positive, there exists a delta positive such that if,

$x \in (1 - δ, 1) \cup (1, 1 + δ) then x + 1.01 \in (2 - ε, 2 + ε)$ .

So, the largest value of $δ$ is given by:

$δ = (2 - 1.01) - 1 δ = 2 - 1.01 - 1 δ = 2 - 2.01 δ = - 0.01$

Since $δ < 0$ , the expression is false.

Step 3. Conclusion.

Therefore $δ < 0$ , the expression is false.

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It is false that $\lim_{x \to 1} (x + 1.01) = 2$ . Express this fact in a mathematical sentence involving $δ$ and $ε$ , to show how the formal definition of limit fails in this case.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Calculation.

Step 3. Conclusion.

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It is false that limx→1x+1.01=2. Express this fact in a mathematical sentence involving δand ε, to show how the formal definition of limit fails in this case.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Calculation.

Step 3. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

Statistics

Discrete Mathematics

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Study anywhere. Anytime. Across all devices.

It is false that $\lim_{x \to 1} (x + 1.01) = 2$ . Express this fact in a mathematical sentence involving $δ$ and $ε$ , to show how the formal definition of limit fails in this case.