Chapter 1: Q. 17 (page 98)

It is false that $\lim_{x \to \infty} \frac{1000}{x} = \infty$ . Express this fact in a mathematical sentence involving $M and N$ , to show how the formal definition of limit fails in this case.

Short Answer

Expert verified

There is some $M > 0$ for which there is no $N > 0$ that would guarantee that if $x > N then \frac{1000}{x} > M$ .

Step by step solution

Step 1. Given information.

The given function is $\lim_{x \to \infty} \frac{1000}{x} = \infty$ .

Step 2. Explanation.

From the given expression, we have, $c = \infty, L = \infty$ .

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

For all $M > 0$ there is some $N > 0$ such that if,

role="math" localid="1649076328357" $x \in (N, \infty) then \frac{1000}{x} \in (M, \infty)$ .

So, the smallest value of $N$ is given by:

$\frac{1000}{x} = M x = \frac{1000}{M}$

Hence, there is some $M > 0$ for which there is no $N > 0$ that would guarantee that if $x > N then \frac{1000}{x} > M$ .

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It is false that $\lim_{x \to \infty} \frac{1000}{x} = \infty$ . Express this fact in a mathematical sentence involving $M and N$ , to show how the formal definition of limit fails in this case.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Explanation.

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It is false that limx→∞1000x=∞. Express this fact in a mathematical sentence involving MandN, to show how the formal definition of limit fails in this case.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Explanation.

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Most popular questions from this chapter

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It is false that $\lim_{x \to \infty} \frac{1000}{x} = \infty$ . Express this fact in a mathematical sentence involving $M and N$ , to show how the formal definition of limit fails in this case.