Chapter 1: Q. 40 (page 107)

For each limit statement , use algebra to find δ > 0 in terms of $ε$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $ε$ .
$\lim_{x \to 1} 2 x^{4} = 2; you may assume δ \leq 1$ .

Short Answer

Expert verified

$δ = m i n (1, \frac{ε}{30})$

Step by step solution

Step1. Given information.

We have been given a limit statement as $\lim_{x \to 1} 2 x^{4} = 2; you may assume δ \leq 1$ .

We have to find $δ in terms of ε$ .

Step 2. Use algebra.

From the given limit statement, we can identify

$f (x) = 2 x^{4} c = 1 L = 2 For ε > 0 |(2 x^{4}) - 2| < ε 2 |x^{4} - 1| < ε |(x^{2} - 1) (x^{2} + 1)| < \frac{ε}{2} |(x - 1) (x + 1) (x^{2} + 1)| < \frac{ε}{2} | x - 1 | \cdot |(x + 1) (x^{2} + 1)| < \frac{ε}{2} δ |(x + 1) (x^{2} + 1)| < \frac{ε}{2}$

Step 3. Put x=2.

$δ \cdot |(2 + 1) (2^{2} + 1)| < \frac{ε}{2} δ \cdot | 3 \cdot 5 | < \frac{ε}{2} δ \cdot | 15 | < \frac{ε}{2} δ < \frac{ε}{30} For 0 < | x - c | < δ, we get | x - 1 | < \frac{ε}{30} Therefore, δ = m i n (1, \frac{ε}{30})$

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For each limit statement , use algebra to find δ > 0 in terms of $ε$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $ε$ .
$\lim_{x \to 1} 2 x^{4} = 2; you may assume δ \leq 1$ .

Short Answer

Step by step solution

Step1. Given information.

Step 2. Use algebra.

Step 3. Put x=2.

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For each limit statement , use algebra to find δ > 0 in terms of ε> 0 so that if 0 < |x − c| < δ, then | f(x) − L| < ε.limx→12x4=2;youmayassumeδ≤1.

Short Answer

Step by step solution

Step1. Given information.

Step 2. Use algebra.

Step 3. Put x=2.

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Mechanics Maths

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Study anywhere. Anytime. Across all devices.

For each limit statement , use algebra to find δ > 0 in terms of $ε$ > 0 so that if 0 < |x − c| < δ, then | f(x) − L| < $ε$ .
$\lim_{x \to 1} 2 x^{4} = 2; you may assume δ \leq 1$ .