Suppose |f(x) − 5|

Question

Suppose |f(x) − 5|<0.1 whenever 00 such that |f(x) − 5|<0.1 whenever 0<|x−2|<δ.

Accepted Answer

Hi, everyone, let's take a look at this practice problem. This problem says given that the absolute value of H Z + 2 is less than 0.2, for all Z in the open interval from 2 to 9, find the range of values for delta greater than 0, ensuring that the absolute value of H of Z + 2 is less than 0.2 whenever 0 is less than the absolute value of Z minus 4, which is less than Delta. We're given 4 possible choices as our answers. For choice A, we have the open interval from 0 to 7. For choice B, we have the open interval from 4 to 9. For choice C, we have the open interval from 0 to 6, and for choice D, we have the open interval from 0 to 2. Now, we're asked to find the range of values for delta greater than 0, ensuring that the absolute value of H of Z + 2 is less than 0.2, whenever 0 is less than the absolute value of Z minus 4, which is less than Delta. So, if we look at that last set of in inequalities, the 0 is less than the absolute value of Z minus 4, which is less than Delta. What that statement is saying is that Z must be within a distance delta from 4. And so that means Z has to be within the open interval from 4 minus delta to 4 plus delta. Now, since Z has to be in this open interval, but we also have that Z has to be in the open interval from 2 to 9, and in both cases, we have the condition that the absolute value of H of Z + 2 is less than 0.2. So we need to have our open interval of 4 minus delta to 4 + delta to be contained within the interval from 2 to 9. And that means for the lower bound, we're going to have 4 minus delta. Has to be greater than 2. And if we solve this for Delta, We see that Delta has to be less than 2. Now, if we look at the upper bounds, we see that 4 plus delta has to be less than 9. And if we solve this for Delta, we see that Delta has to be less than 5. So we now have two conditions on Delta. Delta has to be less than 2, and Delta has to be less than 5. So we're actually going to choose the more restrictive one. The delta less than 2. And we're also told that Delta has to be greater than 0. So that means that 0 has to be less than delta, which has to be less than 2. So that is our range of values for delta. Now, we can actually write this in interval notation, so this is going to be the open interval, going from 0 to 2. And so this is the answer that we were looking for. And if we compare that to our answer choices, the correct answer choice is answer choice D. So, I hope that this explanation has been useful, and I'll see you in the next video.

Suppose |f(x) − 5|<0.1 whenever 0<x<5. Find all values of δ>0 such that |f(x) − 5|<0.1 whenever 0<|x−2|<δ.

Key Concepts

Epsilon-Delta Definition of Limit

Absolute Value Inequalities

Neighborhoods in Calculus

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