(a) Find a number $ \delta $ such that if $ | x - 2 | < \delta $, then $ | 4x - 8 | < \varepsilon $, where $ \varepsilon = 0.1 $.
(b) Repeat part (a) with $ \varepsilon = 0.01 $.
in part A. We are going to find a number delta positive. Such that if the absolute value of x minus two is less than delta then the absolute value of forex minor cities lesson excellent We're absolutely is equal to 0.1. In part B. We repeat what we have done in part A With absolutely equal 0.01. And in fact we are going to do that in general. That is we're going to solve the problem for any value of absence. And what we want to prove here is that we choose at delta in terms of excellent search that this implication here is true, that is absolute value affects ministerial lists and delta will imply that automatically for a cell value for x minus eight is less than absent. So the idea is to choose at delta in terms of absolute um such that these implications here is always true and to do that, we analyze this inequality here. So we start with this After the value of four X -8. We know that this is the actual value of four times X -2 is high parenthesis and the absolute value of the product is the product of the absolute values that this is the absolute value of four times The absolute value of expense to the absolute value of four is 4. So we get four times the absolute value affects -2. And if we had that X managed to in absolute values less than delta then this will be less than for delta. So to have the this expression here which is equal to this chain here. Up to here less than this value And what is going to be Absolutely We just had to take for delta conceptual um or equivalently Delta equals absolute over four. Yeah, so that's the choice we had taken off delta in terms of Excellent. So let's say that this is the way we arrived to the form of delta in terms of Excellent. Where we got to do it. Oh the proof that this is the right choice. We got to do it in a straight way. That is we say like this given positive Absolutely. We take Delta equals absolute over four. Okay then if absolute value affects minus seriously and Delta then The absolute value for X -8 which we have proved already that is equal to four absolute value of x minus delta, X -2. Sorry and x minus choose we're supposed to just listen in absolute value is and delta. So this is lesson for delta and now we say that we have chosen delta this way. So we get this is four times absolutely over four which is absent. So we approved that if this is true Then it's true that this quantity of sort of value for X -8 is less than absolutely. So it's a value for X men S A. Is less than absent. So it means that for any given positive. Absolutely. That's the way to choose delta. And with that election of the value of Delta in terms of Absalom we always have that This inequality implies this other inequality. So we we have proved with this in general that this is the case. That is any value of absolute. We are given, we only have to do this calculation absolute fourth. So For absolutely equal in for a 0.1 we have two shoes, Delta equal absolute 4th equal 0.1 over four. And that is 0.025. And this far A And for absolutely equal 0.01. We have two shoes, Delta equals absolute over four equals 0.01/4. And that's equal to yeah, 0.00 whose life. And that's far be sorry, that's harvey. So we have solved the problem in general. 1st like this and having done the general calculation then we do the particular calculations by putting the values of accident and calculating the violence of correspondent melissa, Delta. Okay. And we uh in some way are talking about the fact that the X goes to two. If X is close to the value to then for expense, for X is close to the value eight, which is true, which is something we can intuitively see that is true.