Q. 27 - Calculus

Question

Use algebra to find the largest possible value of δ or smallest possible value of N that makes each implication true. Then verify and support your answers with labeled graphs.

$\text{ If }x>N\text{, then }\left|\frac{1}{x^2}-0\right|<0.001$

Step-by-Step Solution

Answer

$N=31.62$

Step1. Given information.

We have been given:

$\text{ If }x>N\text{, then }\left|\frac{1}{x^2}-0\right|<0.001$

We have to use algebra to find the largest possible value of δ or smallest possible value of N that makes each implication true.

Step 2. Find the value of N.

We have $\left|\frac{1}{x^2}-0\right|<0.001$.

So we can write:

$$\begin{gathered} \left|\frac{1}{x^2}-0\right|<0.001 \\ \left|\frac{1}{x^2}\right|<0.001 \\ \left|x^2\right|>\frac{1}{0.001} \\ \left|x\right|>\sqrt{1000} \end{gathered}$$

Since $x>N$

$\begin{array}{r}N=\sqrt{1000}\\ \approx 31.62\end{array}$