Q. 9 - Calculus

Question

Use interval notation to fill in the blanks that follow. Your

answers will involve$\delta ,\epsilon , N, \mathrm{and}/\mathrm{or} M$.

If $\underset{x\to \infty }{\lim }f\left(x\right)=2$ then for all $\epsilon >0$there is some$N>0$ such that if $x\in \_\_\_\_\_ \mathrm{then} \in \_\_\_\_\_$.

Step-by-Step Solution

Answer

If $\underset{x\to \infty }{\lim }f\left(x\right)=2$then for all $\epsilon >0$, there is some $N>0$ such that if, $x\in \left(N,\infty \right) \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)\in \left(2-\mathrm{\epsilon },2+\mathrm{\epsilon }\right)$.

Step 1. Given information.

The given function is $\underset{x\to \infty }{\lim }f\left(x\right)=2$.

Step 2. Explanation.

From the given expression, we have, $c=\infty , L=2$.

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

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

$x\in \left(N,\infty \right) \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)\in \left(2-\mathrm{\epsilon },2+\mathrm{\epsilon }\right)$.

Step 3. Statement.

Therefore, statement will be written as: 

If $\underset{x\to \infty }{\lim }f\left(x\right)=2$ then for all $\epsilon >0$, there is some $N>0$such that if, $x\in \left(N,\infty \right) \mathrm{then} \mathrm{f}\left(\mathrm{x}\right)\in \left(2-\mathrm{\epsilon },2+\mathrm{\epsilon }\right)$.