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Question

(a) By graphing the function $f(x)=(\cos 2 x-\cos x) / x^{2}$ and zooming in toward the point where the graph crosses the $y$ -axis, estimate the value of $\lim _{x \rightarrow 0} f(x)$
(b) Check your answer in part (a) by evaluating $f(x)$ for values of $x$ that approach $0 .$

Bobby Barnes · Accepted Answer

they want us to determine the limit. As X approaches, Zero of the function co signed two ex mice Coast on X over X word by first graphing to function as well as evaluating values that are close to X is equal to zero. So I went ahead and did both of those already this chart over here I used Excel to help make, Let&#x27;s see. So approaching from the left, it looks like it&#x27;s getting really close to negative 1.5 and as well. From the right, it looks like it&#x27;s getting really close to negative 1.5. So we might expect that the limit as experts zero of our function would be negative 1.5. And if we were to zoom in on our graph that we had over here, this right here was negative 1.5. So it looks like the Graf also supports what we get from our table here.

Daniel Jaimes · Answer

this is problem number twenty nine Stuart Calculus eighth edition. Section two point two Party By grafting the function at FedEx is equal to the quantity of co sign two. X Man is cosign of eggs divided by X squared and zooming in towards the point where the graph crosses the Y axis has to meet. The value of the limit is X approaches. Zero for the function X. We&#x27;re going to take this function cosign of two x minus cosign of X. That whole quantity, the better. Right X squared. And we&#x27;re going to use a graphing calculator or, in his case, a Griffin spreadsheet. And we&#x27;re going to investigate what the behavior of the function is as we approach X equals zero from the left and from a rating. So here&#x27;s a plot of the function. We&#x27;ve already zoomed in very closely. Two. The area of X equals zero, which is this line as we a closer from the left, the function of purchase a value of about negative one point five. As we approach from the right, we also purchase value that&#x27;s close to negative one point five. And so our first estimate for party Yeah, that we come from graphically is that the limit is equal to negative one point five. We&#x27;re gonna check this answer and party by evaluating the function for values of X that approach zero. And if we go back to a spreadsheet, you see that this plot was actually made using values that we&#x27;re very close to zero and we will focus specifically on the table of values and show that from the right hand side as we hear closer and closer, zero from the right, we see that the values got closer and closer to negative one point five Similarly, from the left values that are negative and close to zero, we see the calculations getting closer and closer to about negative one point five Toy reached this point. At which point we can say defendant Arlene, that power lim as explosions Zero for this function is negative. One point five and we have confirmed too pretty

Daniel Jaimes · Answer

This is problem number forty six of the Stuart Calculus eighth edition Section two point two Partying By grabbing the function improve, X equals ten jina for X the Betamax and zooming in toward the point where the graph crosses the Y axis. I estimate the value of Element and six approaches zero of this function. F Riddick E graphing tool and plug in this function tender for X rex, and we zoom in very closely two x zero. We see that the function approaches for from the left, and it also purchase for from the right. And although dysfunction is undefined at X equals zero because that would make the denominator equal to zero, we see that the limit is definitely equal two four because the limit from the left equals limit from the right equals four Carpi take care. Answered party by evaluating F for values of X that approach zero. So we&#x27;re going on or for Tara spreadsheet and we&#x27;LL see that we used a table of values in order for us to plant dysfunction and as we see approaching the function approaching X equals zero from the right kidding X values closer and closer. Zero. We see that the function approaches Thor closer and closer. And also this is true from the left. As we approach the arrow from the left, the function gets closer and closer for in this confirms our solution at the limit has experts. Zero of the function F is equal to four.

Paul George · Answer

everybody we&#x27;re doing Chapter two, Section three Problem 28. We are doing the limit as X approaches. Zero of sign of X over Sign Pi X. So if I just quickly, you know, draw it out, at least for you know what it looks like? The reason why this is such a problem because you are trying to find as X approaches zero. Then if you plug in zero, right, If we&#x27;re talking about limits, we&#x27;re tryingto find out trends as we get to zero. But if you&#x27;re trying to think about like Okay, well, what is it? A zero? You plug in zero for this, you get sign of zero, which is a row. You get signed of pi times zero, which is also zero. So you get zero divided by zero. So if you divide something by zero, it&#x27;s undefined. If you take zero divided by something, it&#x27;s zero. So you get zero divided by something. It&#x27;s undefined. You get an unknown quantity. It&#x27;s this very weird thing that can happen. So that&#x27;s why this ends up being a really interesting problem. The cool thing about it is, that part is is saying, Hey, graft this thing right, and then look at it. And so, if you if you pull out your favorite graphing, you know, device, whatever you happen to have, um you kind of zoom in on this thing, Then you&#x27;ll end up finding that right here on the axis, you end up actually getting this thing that looks quite a bit. Um, like if you had just have, like, one over sign, which is CO c can. So the nice thing about this is that technically, at this point, it is that undefined. But there&#x27;s this really cool thing is that you can actually take, um, you can take an approximation to get pretty close. And so if you zoom in, you know, pretty far you get something that looks about like 0.31 ish, which is pretty close. And so that&#x27;s you can you can look at that and you might go Well, what is what is this 0.31 coming from? And so that&#x27;s kind of like what part they want you to do, right? Look at this graph. Kind of see what&#x27;s going on, and you end up getting something likewise your one and then and then investigated. Try to find out what&#x27;s going on, which is what part B is so part be asked to you to kind of like, Hey, take your take your ah little walk down a number line rights and I want you. You know, you said that the 0.31 was in there right at zero. And these are X values. And then, you know, these are like our limits stuff, right? And then if you plug in stuff that&#x27;s close, then this is sort of what what the game is. You put it in a 0.5, you plug in a 0.1, you get some numbers. You know, you might get like point for seven, and then you get like 70.38 and you get closer and closer and closer, and you you get something that feels a little bit like 0.31 Here&#x27;s the interesting thing about this is that you might be wondering, Where is this 0.31 coming from? Well, it&#x27;s much more than 0.31 It&#x27;s actually if you take on, we get more into this and calculus, but actually, if you take this one from inside of the sign and the pie from inside of the sign and you get one divided by pi. One divided by pi is very close to 10.31 and that is not a coincidence. That&#x27;s not exactly what this problem is asking you, but I think it&#x27;s illuminating as to what you&#x27;re going to be doing pretty soon. In the world of calculus, once we are done with limits, I hope this was helpful right until that time.

Daniel Jaimes · Answer

This is problem number thirty of Stuart Calculus, eighth edition, Section two point two. Party has to meet. The value of the limit is experts zero of sign of X threated by sign. Ah, pi X, I graphing the function of state Your answer correct to two decimal, please. So we are going to take this function sign of X divided by sign of pranks. And we&#x27;re going to plant this function around this area. X goes to zero. You do this with your graphing calculator or any other graphing device here in our spreadsheet. We have set up this plot of this function and we assumed an already very close to zero. And now we&#x27;re preserving exactly where this my cross, the y axis. You see that it approaches from the left in a hurry. That purchase value approximately zero point three two. But it may be more Kurt to say, a value, lest then zero point three two. However, using a graph is not the most direct answer. So for the time being for party, we&#x27;re going to estimate this. Yes, we can and say, that is, um it is approximately zero plane three two for part B we&#x27;re going to check her answer and evaluate the function for various values that are close to zero. And in this way we may be able to estimate the limit further in a spreadsheet. We have used some values to plot the function here, and we&#x27;re gonna take a look at this list of values and determined what the limit might be. We have values here that approach zero from the right, and we see that the values decrease closer and closer to a number. Their point three one eight three Ah, and so on. Same thing from the left hand side, we get about the same values. All the calculations are mirrored about to this y axis. And finally, what we can conclude is that the actual limited is closer to about zero point three one eight, which is a very close to our estimation of zero point three two term party. We&#x27;re going to say that the limit is actually approximately zero point three one eight, which is a slight Lee better estimate than party

Daniel Jaimes · Answer

This is prime number twelve, the sewer calculus, eighth division, section two point six party use. A graph of function F is equal to quantity. One minus two of rex, razed to the X power to estimate the value of the limit has exported divinity of this function after correct two decimal places. So if we are, we&#x27;re deployment this function one minus two wrecks that quantity race to the X power. We will get this function here where it would level off, had a certain value. And if we were to trace along dysfunction and see what value that was equal to, we see that its approach is approximately, sir point one three five on DSO. We would say that this limit is approximately zero point one four two two decimal places and that&#x27;s our answer from using a graph. Isn&#x27;t stable values for this function as to meet the limited for two small places? So with table vise, we should be able to choose a numbers large enough that we can see the trend of the function and how much it decreases by and where it seems to be approaching for very, very large numbers. We see that this function approaches approximately point one three five, three, two, two, four two Ford Decimal places on. And so we would say that our limit is more accurately equal to zero point one three five, and that is our final answer.

Problem

(a) Estimate the value of $\quad \lim _{x \righta…

Problem 27 Easy Difficulty

(a) By graphing the function $f(x)=(\cos 2 x-\cos x) / x^{2}$ and zooming in toward the point where the graph crosses the $y$ -axis, estimate the value of $\lim _{x \rightarrow 0} f(x)$
(b) Check your answer in part (a) by evaluating $f(x)$ for values of $x$ that approach $0 .$

Answer

(a) $\lim _{x \rightarrow 0} f(x)=-1.5$
(b) $\lim _{x \rightarrow 0} f(x)=-1.5$

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James Stewart

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Catherine R.

Anna Marie V.

Samuel H.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

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(a) $\lim _{x \rightarrow 0} f(x)=-1.5$(b) $\lim _{x \rightarrow 0} f(x)=-1.5$

Biocalculus Calculus for the Life Sciences

Catherine R.

Anna Marie V.

Samuel H.

Joseph L.

Precalculus Review - Intro

Functions on the Real Line - Overview

James StewartBiocalculus Calculus for the Life Sciences

Catherine R.

Anna Marie V.

Samuel H.

Joseph L.

(a) $\lim _{x \rightarrow 0} f(x)=-1.5$
(b) $\lim _{x \rightarrow 0} f(x)=-1.5$

James Stewart

Biocalculus Calculus for the Life Sciences