Use the given graph of f to find a number δ such that if |x−1|&lt;δ then |f(x)−1|&lt;0.2 δ=?

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Use the given graph of f to find a number δ such that if |x−1|&amp;lt;δ then |f(x)−1|&amp;lt;0.2 δ=?

curwyrm · Accepted Answer

Step 1 Given: |x−1|&amp;lt;δ then |f(x)−1|&amp;lt;0.2 To find the number δ then is to find how close x has to be to 1. In order for f(x)&amp;lt;0.2 that is close to 1. or, How close does x have to be to 1 (on either side), for f(x) to be between 0.8 and 1.2? So, by the graph: It is to clear that on the left side of x=1, x can be within 0.3, but on the right side, it&#039;d have to be within 0.1, of 1, for f(x) to be that close to 1. Now, f(x) is within 0.2 of 1 ( which is what |f(x)−1|&amp;lt;0.2 is saying), we take 0.1, which is sure to work on both sides. Hence, δ=0.1 Here used absolute values around each difference to show that the difference would work on both sides: a positive or negative difference would come out the same. When we take the absolute values, Looks like it&#039;s leading up to understanding &quot;derivatives&quot; or instantaneous slopes.

Jeffrey Jordon · Accepted Answer

Explanation:Seriously, you&#039;ll like this.The red horizontal segments are distance 0.2 away from 1. These are&amp;nbsp;y=1-ϵ&amp;nbsp;and&amp;nbsp;y=1+ϵ.The corresponding red vertical segments are atx=1−0.3&amp;nbsp;and&amp;nbsp;x=1+0.1Choose&amp;nbsp;δ=0.1&amp;nbsp;-- the smaller distance from the limit point. Then whenever0&amp;lt;|x−1|&amp;lt;δ,&amp;nbsp;f(x)&amp;nbsp;will be between the red horizontal lines.In other words,&amp;nbsp;|f(x)−1∣&amp;lt;ϵ.

Jeffrey Jordon · Accepted Answer

Explanation:  Seriously, you&#039;ll like this. The red horizontal segments are distance 0.2 away from 1. These are y=1−ϵ and y=1+ϵ. The corresponding red vertical segments are at x=1−0.3 and x=1+0.1 Choose δ=0.1 -- the smaller distance from the limit point. Then whenever 0&amp;lt;|x−1|&amp;lt;δ, f(x) will be between the red horizontal lines. In other words, |f(x)−1∣&amp;lt;ϵ

Use the given graph of f to find a number \delta such that if |x-1|<\delta then |f(x)-1|<0.2\delta=?

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