Chapter 1: Q. 51 (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 localid="1648028930198" $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$ .
localid="1648029006873" $\lim_{x \to 3} (2 - x^{2}) = - 7, ε = 0.01$

Short Answer

Expert verified

The largest value of $δ \approx 0.00167$ .

Step by step solution

Step 1. Given Information.

The given expression is $\lim_{x \to 3} (2 - x^{2}) = - 7, ε = 0.01$ .

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

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

Now, the largest value of delta is given by,

localid="1648202643668" $2 - x^{2} = L + ε 2 - x^{2} = - 7 - 0.01 2 - x^{2} = - 7.01 2 + 7.01 = x^{2} 9.01 = x^{2} x = \sqrt{9.01}$

Thus,

localid="1648202678785" $δ = \sqrt{9.01} - 3 δ \approx 0.00167$

Step 3. Conclusion.

The largest value of $δ \approx 0.00167$ .

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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 localid="1648028930198" x∈(c-δ,c)∪(c,c+δ)thenf(x)∈(L-ε,L+ε).localid="1648029006873" limx→32-x2=-7,ε=0.01

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Conclusion.

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