Chapter 1: Q. 49 (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="1648028606662" $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$ .
localid="1648028619268" $\lim_{x \to 0} \frac{1 - \cos (x)}{x} = 0, ε = \frac{1}{2}$

Short Answer

Expert verified

The largest value of $δ \approx 1.11$ .

Step by step solution

Step 1. Given Information.

The given expression is $\lim_{x \to 0} \frac{1 - \cos (x)}{x} = 0, ε = \frac{1}{2}$ .

Step 2. Explanation.

From the given expression, we have, $c = 0, L = 0$

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

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

$x \in (0 - δ, 0) \cup (0, 0 + δ) then \frac{1 - \cos (x)}{x} \in (0 - ε, 0 + ε)$

Now, the largest value of delta is given by,

localid="1648206479588" $\frac{1 - \cos (x)}{x} = L + ε \frac{1 - \cos (x)}{x} = 0 + \frac{1}{2} \frac{1 - \cos (x)}{x} = \frac{1}{2} x = 2 - 2 \cos (x) x \approx 1.11$

Thus,

$δ = 1.11 - 0 δ \approx 1.11$

Step 3. Conclusion.

The largest value of $δ \approx 1.11$

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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 ofδsuch that if localid="1648028606662" x∈(c-δ,c)∪(c,c+δ)thenf(x)∈(L-ε,L+ε).localid="1648028619268" limx→01-cosxx=0,ε=12

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

Step 3. Conclusion.

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