Question

use the given graph of (f(x)=sqrt{x}) to find a number (delta) such that if (|x - 4|ltdelta) then (|sqrt{x}-2|lt0.4).

Explanation:

Step1: Solve the inequality $\vert\sqrt{x}-2\vert < 0.4$

We have two - cases:
Case 1: $\sqrt{x}-2 < 0.4$, then $\sqrt{x}<2 + 0.4=2.4$, and $x < 2.4^{2}=5.76$.
Case 2: $-(\sqrt{x}-2)<0.4$, which simplifies to $\sqrt{x}-2>-0.4$, then $\sqrt{x}>2 - 0.4 = 1.6$, and $x>1.6^{2}=2.56$.

Step2: Find the distance from $x = 4$

We want to find $\delta$ such that if $\vert x - 4\vert<\delta$, then $\vert\sqrt{x}-2\vert < 0.4$.
The distance from $x = 4$ to $x = 2.56$ is $\vert4 - 2.56\vert=1.44$.
The distance from $x = 4$ to $x = 5.76$ is $\vert4 - 5.76\vert = 1.76$.
We take the smaller of these two distances.

Answer:

$1.44$