Q. 24 - Calculus

Question

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

If $0<\left|x-3\right|<\delta$, then $\left|\frac{1}{x}-\frac{1}{3}\right|<0.2$

Step-by-Step Solution

Answer

The required value of N is $\left|x-3\right|<0.6x$.

The required graph is given below,

Step 1. Given information.

Consider the given question,

If $0<\left|x-3\right|<\delta$, then $\left|\frac{1}{x}-\frac{1}{3}\right|<0.2$

Step 2. Find the value of &#948; .

It can be written,

$$\begin{gathered} \left|\frac{3-x}{3x}\right|<0.2 \\ \left|\frac{x-3}{3x}\right|<0.2 \\ \left|x-3\right|<0.2·3x \\ \left|x-3\right|<0.6x \end{gathered}$$

Step 3. Sketching the function to illustrate the choice of &#948; .

Sketching the function, we get,