Q. 7 - 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 $\underset{x\to 2}{\lim }f\left(x\right)=5$then for all $\epsilon >0$there is some $\delta >0$ such that if $x\in \_\_\_\_\_ \mathrm{then} f\left(x\right) \in \_\_\_\_\_$.

Step-by-Step Solution

Answer

If $\underset{x\to 2}{\lim }f\left(x\right)=5$ then for all epsilon positive, there exists a delta positive such that if, $x\in (2-\delta ,2)\cup (2,2+\delta ) \mathrm{Then} f\left(x\right)\in (5-\mathrm{\epsilon },5+\mathrm{\epsilon })$.

Step 1. Given information.

The given function is $\underset{x\to 2}{\lim }f\left(x\right)=5$.

Step 2. Explanation.

From the given expression, we have, $c=2, L=5$.

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

For all epsilon positive, there exists a delta positive such that if,

$x\in (2-\delta ,2)\cup (2,2+\delta ) \mathrm{Then} f\left(x\right)\in (5-\mathrm{\epsilon },5+\mathrm{\epsilon })$.

Step 3. Statement.

Therefore, statement will be written as:

If $\underset{x\to 2}{\lim }f\left(x\right)=5$ then for all epsilon positive, there exists a delta positive such that if, $x\in (2-\delta ,2)\cup (2,2+\delta ) \mathrm{Then} f\left(x\right)\in (5-\mathrm{\epsilon },5+\mathrm{\epsilon })$.