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Problem 50 Hard Difficulty

(a) Evaluate $ h(x) = (\tan x - x)/x^3 $ for $ x $ = 1, 0.5, 0.1, 0.05, 0.01, and 0.005.

(b) Guess the value of $ \displaystyle \lim_{x \to 0}\frac{\tan x - x}{x^3} $.

(c) Evaluate $ h(x) $ for successively smaller values of $ x $ until you finally reach a value of 0 for $ h(x) $. Are you still confident that your guess in part (b) is correct? Explain why you eventually obtained 0 values. (In Section 4.4 a method of evaluating this limit will be explained.)

(d) Graph the function $ h $ in the viewing rectangle $ [-1, 1] $ by $ [0, 1] $. Then zoom in toward the point where the graph crosses the $ y $ -axis to estimate the limit of $ h(x) $ as $ x $ approaches 0. Continue to zoom in until you observe distortions in the graph of $ h $. Compare with the results of part (c).

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we're going to evaluate the function H of X equals tangent of x minus X. And that over xQ. For the values of X equal to 10.50 point 10.50 point 01 and 0.5. That's part A. And part B. We guess The value of the limit when H. Sorrow and eggs goes goes to zero. Here's eggs exports terror of age of Next. And the idea is that in part A. We are evaluating the function age at smaller values of X. That is valleys are each time closer to zero. And with that birds are going to try to guess the value of that limit and birth C. With about age for smaller and smaller values of eggs. That is we go down this value here in order to see what happened to the value of age. And we are going to see that we finally are going to reach the about zero and we try to explain why that happens. So part A. We make a table of values of age so get table X. Age And even at 1 0.1. Our ceo five first yes 05. Yeah That's your .1 then 0.05 Than 0.01. And finally 0.005. So that's it. That's the father's will get to even wait function. And we have done that. You've seen my club And we got the following results at one. We get 0.55 seven 4 6 or eight. You seen six decimals At 0.5. 0.37 04 20. And that's Europe in one we got zero 33 4672. Then to your point 33 3 again 667 In a 2.01 regards zero point And we have our four times 3 and then 47. And finally at 0.005 we got zero. Let. Let me see how many threes we have want five, five threes and then seven. And as we can see here but we can guess for the lemon. Looking at these values in this table Is that we're going to get 0.3 period. That is finally many three symbol. This is the case. We guess that limit when x goes to zero of age Must be zero three period which is equal to 1/3. That is you've heard that but it's true that we haven't put Values so small here because we stopped at 3.005 but we can do then 3.001. And smaller than that in order to be confident that this will be the the limit. So in part C cannot show that's exactly here. But it's true that if you keep them either awaiting the function at smaller values of eggs, a thing is going to happen for example and say that You descend up to zero zero She was here one, you are still looking at the sprint here there is have more values three in the decimal part that are and That enforced reinforced the idea that the limit is 1/3. Yeah. If you put a smaller value than something being to happen is well it's too small. For example it More serious here for example 0.000001. At that number the quantity here, the number of three in the national park start to um to get to decrease that is we will have less number three dozen part example. It's like the number is increasing some way and if we do that or more even more violent tyvek stand we will have eventually zero for temple in this case I get a zero with mhm um with seven 10 to the negative page. That is seven Zeros with this number I get age equal serum. That's true that this situation can be different in another system. And use the med lab which works in double precision that is We have 15 decim of precision in the decimal results. And that means that maybe in another computer or in a calculator. This happens before that is force a number greater than this. We have already a zero forage and then we have that sensation of the idea that the images serum That is. We start to not being confident to the result we had in this table. That is the gas of 1/3 And the truth is the truth is that the limit is 1/3. But we're going to show up and and in fact showing what happened, We're going to prove that limit is 1/3 in some way. So what we're gonna do here, he's a following first. Why this happened? Then we have the expression tangent of X minus X over execute. So we have a difference subtraction in the numerator. And we're going to do a simple calculation here. Let's say let's calculate the tangent line two 10 that the function tangent of eggs at zero for that. We need the derivative of attention to facts. You know that the derivative of tangent of eggs its second square of X. So the derivative of the tangent of X evaluated at X equals zero is second square zero. Remember that that's equal to one over co sign square of zira consent of serious one. So that we get one here. So this slow of detention line To the function tangent of X at zero is 1. And we know that the tangent of 00. So the line mhm right buses mm hmm through The zero which is the value of X. And its image through the function tended. Okay. So if the slope is one and the line passes from 00. The equation of that tangent two Tangent of X zero. I get why zero equals slope one times X zero. That's because it passes through 00 And has slow one that give us why it was X. That is the identity. So the tangent line to the tangent function that zero is the identity function. It means that withdraw the tangent or the first branch of the tangent which is defined between negative. I have open and I have open here at zero, withdraw the tangent line and it will be the identity line. It means that closed zero. When we get closer and closer to zero. The function tangent of X is very similar to X. Remember dad, the change in line close to the point of tendency. It's a very good approximation or the authorization to the function that is we can say here that tangent of X. It's very similar to X when X these clothes to Syria. And it means, yeah, the difference produces cancellation cancellation. That is the quantities as very similar close to zero. And so we had we lost accuracy because we have this obstruction of two very similar numbers. And so what we get in fact is numbers of the Sort of zero points. Here's here's here's something and if we keep on doing that, it becomes a moment when the difference seems to be zero. But it's not serious. Just we are losing accuracy. That's what we call a numerically cancellation. Yeah, okay. And for that reason cancellation and for that reason we get to do that operation all the way around in order to avoid cancellation. And what we can do here to avoid cancellation in that numerator in that term is to for example develop the taylor pulled amir of tangent of X, not of a high so high order birth, moderate order. And with that we can represent the function, attention to effects through a polynomial and not through this function itself. So in this case again, but remember the tailor production is very good or is very similar to the functions a good approximation to the function near the point of development of the pollen almond. So we consider taylor Well, you know, I'm real of the tangent of X. Uh develop develop uh zero off for example, order or degree that say five for example, I'm not going to go through the details of calculation because it's another thing but can do it is lived through the calculations of the derivatives of The tangent function up to the degree we want. In this case I'm going to use 35. I would know that that polynomial P five of X is equal to X plus X. Cube over three plus uh two X to the fifth over 15. That's people who don't know Kilburn Annual of Degree five develop at ankles here. You know that For eggs close to zero. P5 of X is a good approximation of the change into vets. That's true general. There is given a function with some derivatives or all derivatives that we want then we can say that near the point of development where we developed the polynomial. The function and the pollen um are very close. That's not true. If we go away from the point of development of the economic, if we are close to the point of development of the tale of Britain normal we are very close to the function Near that point in this case near zero. This will never get to be close to this function and we can use it instead of the function that it means. And so we can say that The tangent of X -X is approximately two P five of X minus six. But look at this this expression if we do this operation here we get Passing these X to the left is eggs Cube over three Plus two eggs to the 5/15. And we when we divide that by three, that is tangent of X divided by x cubed or which is the limit. We are calculated attention to vex minus X over X. Q. That is opportunity equal to X cube over three plus two eggs to the fifth over 15th. All that over execute we distribute The Numerator two terms in the numerator With the denominator and we had 1/3 because X cubed cancel out plus two eggs Square because five ministries to over 15. And this way we can say that limit when X goes to zero growth of tangent of X minus X of x cubed. It's about the same of the limit when X goes to zero of one third plus two eggs square over 15 And Exhale 20. This term is no but this is a constant. So we get one third which is the limit. We have guests in perpetuity. So that's a limit. And it's important to notice that this well a normal here which we are using instead of the function has no consolation anymore because there are only sons here if X is negative because all the powers are or we will have again and some and not any difference. So there's no more cancellation here. And so we have calculated limit currently. Yes in the alternative representation of the function tangent of X. And that happens in the some other problems the same way. But in another case is what we got to do is to transform the geometrical function using some terra metric identity. In this case it's terrible. Norick solves the problem. So we have that this is the limit. Another way to find this limit because the numerator goes to zero. It is if we evaluate ears to the numerator, we get zero attention to zero minus 00. And the denominator When X00 also. And so we can use what we call love to rule. But this implies using derivatives. That is we in this case we know that this Because both numerator and denominator separately goes to zero and X goes to zero in this case. Then we can say that this limit is equal to The limit when X goes to zero off the derivative respect to eggs of the narrator, over the derivative respect to X of the denominator. That is Limit when X goes to zero off 2nd square of eggs minus one over three eggs square. If we either late again at zero we get enumerators zero because he can square series one. So it gets one minutes, 10. And determination we get again zero because 30 square zero. And we get to apply the same rule again. And we do that until we get a number. So again we say this here um she can square Of X -1 over three X squared derivative of that. It's important to notice that this little rule uh is calculating the derivatives separately. It's not the derivative of the Kocian. Is there? When we have zero limit zero and the numerator limit zero either denominator, we find the derivatives of numerator and denominator separately. And that new function we take the limit again. And if we again get zero in the numerator and zero in the denominator we apply the rule again. So here that give us the limit when exposed to zero of two seconds of x times purity of second, the second time standing. So we get this over six eggs. Now we get zero again because we get steroids Liberator and two times one times zero in the numerator. So we get 000 and zero in both terms. So we get again and play the rule derivative of the numerator. Okay. Over the derivative of the denominator. And now we are close to a result because um We have in the denominator six and no more eggs in the denominator and numerator we have a product. So we get up like the derivative of product to is a constant two times. So we get the derivative of second square times tangent of X. So the everything. Second square is two second bags times already of C. Can't is second tangent. So his second squared times tangent. And that times tangent. That is these expressions only the derivative of secret square. Sorry, here is seeking to the one of course is two times the same function to the one times the derivative of seeking that is second mm uh, time standards. Okay, it's correct. So we get that because we have this because the derivative and these square because we multiply by the other function tends intimates. Then we have plus he can square of eggs. And the relative to tangent. Seacon square. So we get second to the fourth. So here is 2nd. That's a good news. She can't To the 4th. That's good news because we don't have tangent to this term. So we can't even wait. This limit by evil waiting X equals zero. This first term is zero because we have tangent of X intends to tangent of 00. But this term here is one Because it's one over goes into the 4th and the co sign a series one. So we get to over six. Very good. That is 1/3. So we get the same result. But applying this time the local to rule For a limit of the form zero over sue. But this idea is very interesting because we can solve many problems using taylor expansions or taylor renominates like we did here. So the summary is that the calculations can mislead the conclusion because we have some sort of numerical problem and we get to be very careful that and some problems we can be surprised by the fact that there is medical results. Has no so, so much sense or are misleading at least. So we get to be careful if we have differences of values or something like that. If we are cancellation or maybe we have rounding problems in some occasions. So this is the solution of the given up