Provide an appropriate response.
- Given the interval $( a , b )$ on the $x$ -axis with the point $x _ { 0 }$ inside, find the greatest value for $\delta > 0$ such that for all $x , 0 <$ $\left| x - x _ { 0 } \right| < \delta \Rightarrow a < x < b \text {. }$
a = 9, b = 19, x0 = 16

Question

Provide an appropriate response.
- Given the interval $( a , b )$ on the $x$-axis with the point $x _ { 0 }$ inside, find the greatest value for $\delta > 0$ such that for all $x , 0 <$ $\left| x - x _ { 0 } \right| < \delta \Rightarrow a < x < b \text {. }$
a = 9, b = 19, x0 = 16
A) 1
B) 3
C) 4
D) 7

Quizplus · Accepted Answer

The greatest value for $\delta$ is determined by the distance from $x_0$ to the nearest endpoint of the interval $(a, b)$. Since $x_0 = 16$, the distances are $16 - 9 = 7$ and $19 - 16 = 3$. The smallest of these is 3, so $\delta = 3$.

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