Chapter 1: Q. 52 (page 97)

For each limit $\underset{x \to c}{l i m} f (x) = L$ in Exercises 43–54, use graphs and algebra to approximate the largest value of $δ$ such that if $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$ .
$\lim_{x \to 3} (2 - x^{2}) = - 7, ε = 0.001$

Short Answer

Expert verified

The largest value of $δ = 0.0001666$ .

Step by step solution

Step 1. Given Information.

The given expression is $\lim_{x \to 3} (2 - x^{2}) = - 7, ε = 0.001$ .

Step 2. Explanation.

From the given expression, we have, $c = 3, L = - 7$ .

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

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

role="math" localid="1648189035483" $x \in (3 - δ, 3) \cup (3, 3 + δ) then (2 - x^{2}) \in (- 7 - ε, - 7 + ε)$

Now, the largest value of delta is given by,

$2 - x^{2} = L + ε 2 - x^{2} = - 7 - 0.001 2 - x^{2} = - 7.001 2 + 7.001 = x^{2} 9.001 = x^{2} x = \sqrt{9.001}$

Thus,

$δ = \sqrt{9.001} - 3 δ \approx 0.0001666$

Step 3. Conclusion.

The largest value of $δ \approx 0.0001666$ .

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For each limit $\underset{x \to c}{l i m} f (x) = L$ in Exercises 43–54, use graphs and algebra to approximate the largest value of $δ$ such that if $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$ .
$\lim_{x \to 3} (2 - x^{2}) = - 7, ε = 0.001$

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Conclusion.

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For each limit limx→cf(x)=Lin Exercises 43–54, use graphs and algebra to approximate the largest value of δsuch that if x∈(c-δ,c)∪(c,c+δ)thenf(x)∈(L-ε,L+ε).limx→32-x2=-7,ε=0.001

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Conclusion.

One App. One Place for Learning.

Most popular questions from this chapter

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For each limit $\underset{x \to c}{l i m} f (x) = L$ in Exercises 43–54, use graphs and algebra to approximate the largest value of $δ$ such that if $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$ .
$\lim_{x \to 3} (2 - x^{2}) = - 7, ε = 0.001$