Chapter 1: Q. 50 (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 $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$ .
$\lim_{x \to 0} \frac{1 - \cos (x)}{x} = 0, ε = \frac{1}{4}$

Short Answer

Expert verified

The largest value of $δ \approx 0.511$ .

Step by step solution

Step 1. Given Information.

The given expression is $\lim_{x \to 0} \frac{1 - \cos (x)}{x} = 0, ε = \frac{1}{4}$ .

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,

$\frac{1 - \cos (x)}{x} = L + ε \frac{1 - \cos (x)}{x} = 0 + \frac{1}{4} \frac{1 - \cos (x)}{x} = \frac{1}{4} x = 4 - 4 \cos (x) x \approx 0.511$

Thus,

$δ = 0.511 - 0 δ \approx 0.511$

Step 3. Conclusion.

The largest value of $δ \approx 0.511$ .

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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 $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$ .
$\lim_{x \to 0} \frac{1 - \cos (x)}{x} = 0, ε = \frac{1}{4}$

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→01-cosxx=0,ε=14

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 $x \in (c - δ, c) \cup (c, c + δ) then f (x) \in (L - ε, L + ε)$ .
$\lim_{x \to 0} \frac{1 - \cos (x)}{x} = 0, ε = \frac{1}{4}$