Q. 17 - Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1. Given information.

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

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,

$x\in \left(N,\infty \right) \mathrm{then} \frac{1000}{\mathrm{x}}\in \left(\mathrm{M},\infty \right)$.

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

$$\begin{gathered} \frac{1000}{x}=M \\ x=\frac{1000}{M} \end{gathered}$$

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