Chapter 1: Q. 42 (page 107)

For each limit statement in Exercises $41 - 44$ , use algebra to find $δ$ or $N$ in terms of $ε$ or $M$ , according to the appropriate formal limit definition.
$\lim_{x \to \infty} \frac{x - 1}{x} = 1$ , find $N$ in terms of $ε$ .

Short Answer

Expert verified

For $\lim_{x \to \infty} \frac{x - 1}{x} = 1$ , $N = \frac{1}{ε}$ .

Step by step solution

Step 1. Given information

$\lim_{x \to \infty} \frac{x - 1}{x} = 1$ .

Step 2. From the limit expression,

$f (x) = \frac{x - 1}{x} c = \infty L = 1$

Now for localid="1648021305175" $ε > 0$ .

localid="1648021309449" $|(\frac{x - 1}{x}) - 1| = |\frac{x - 1 - x}{x}| = |\frac{- 1}{x}| = \frac{1}{|x|} < \frac{1}{N} = ε$

Therefore, for $x > N$ , we get localid="1648021314200" $|(\frac{x - 1}{x}) - 1| < ε$ .

Hence,localid="1648021320692" $N = \frac{1}{ε}$ .

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For each limit statement in Exercises $41 - 44$ , use algebra to find $δ$ or $N$ in terms of $ε$ or $M$ , according to the appropriate formal limit definition.
$\lim_{x \to \infty} \frac{x - 1}{x} = 1$ , find $N$ in terms of $ε$ .

Short Answer

Step by step solution

Step 1. Given information

Step 2. From the limit expression,

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For each limit statement in Exercises 41-44, use algebra to find δor Nin terms of εor M, according to the appropriate formal limit definition.limx→∞x-1x=1, findNin terms ofε.

Short Answer

Step by step solution

Step 1. Given information

Step 2. From the limit expression,

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Statistics

Logic and Functions

Pure Maths

Decision Maths

Probability and Statistics

Study anywhere. Anytime. Across all devices.

For each limit statement in Exercises $41 - 44$ , use algebra to find $δ$ or $N$ in terms of $ε$ or $M$ , according to the appropriate formal limit definition.
$\lim_{x \to \infty} \frac{x - 1}{x} = 1$ , find $N$ in terms of $ε$ .