Q. 25 - 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 $x\in \left(1,1+\delta \right)$, then $\left|\sqrt{x-1}-0\right|<0.5$

Step-by-Step Solution

Answer

The required value of N is $\delta =0.25$.

The required graph is given below,

Step 1. Given information.

Consider the given question,

If $x\in \left(1,1+\delta \right)$, then $\left|\sqrt{x-1}-0\right|<0.5$.

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

Write the interval notation $x\in \left(1,1+\delta \right)$ in inequality first:

$1<x<1+\delta$

Subtract $1$ from every side,

$$\begin{gathered} 1-1<x-1<1+\delta -1 \\ 0<\left|x-1\right|<\delta \end{gathered}$$

From $\left|\sqrt{x-1}-0\right|<0.5$, we can write,

$$\begin{gathered} \left|\sqrt{x-1}-0\right|<0.5 \\ \left|\sqrt{x-1}\right|<0.5 \\ \left|x-1\right|<{\left(0.5\right)}^2 \\ \left|x-1\right|<0.25 \end{gathered}$$

As $0<\left|x-1\right|<\delta$, then,

$\delta =0.25$

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

Sketching the function, we get,