killjoy1990xb9

2022-01-07

Use the given graph off to find a number $\delta$ such that if $|x-1|<\delta$ then $|f\left(x\right)-1|<0.2$

Philip Williams

Beginner2022-01-08Added 39 answers

$\left|X\right|<a$ implies $-a<X<a$

Simplify by adding 1 to each

$|f\left(x\right)-1|<0.2$

$-0.2<f\left(x\right)-1<0.2$

$0.8<f\left(x\right)<1.2$

The graph shows that if $0.8<f\left(x\right)<1.2$ x must verify $0.7<x<1.1$

On the other hand, x is required to confirm this inequality:

$|x-1|<\delta$

$-\delta <x-1<\delta$

If we can find $\delta$ that verify $-0.3\le -\delta <x-1<\delta \le 0.1$, we can make sure that $0.7<x<1.1$

so $\delta$ must verify two conditions: $\delta \le 0.1$ and $-0.3\le -\delta$

which means that $\delta \le 0.1$ and $0.3\ge \delta$

$0.7<x<1.1$

$0.7-1<x-1<1.1-1$

$-0.3<x-1<0.1$

We can choose $\delta =0.1$, so to be sure two conditions are satisfied, but if you want, you can choose any value.

Toni Scott

Beginner2022-01-09Added 32 answers

You can find it using the graph

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