find-the-limits-lim-_x-rightarrow-0-fractan-3-x2sin-2-5-xx2

Question

Find the limits.
$$
\lim _{x ightarrow 0} \frac{	an 3 x^{2}+\sin ^{2} 5 x}{x^{2}}
$$

R M · Accepted Answer

Brooklyn. Number 40. The limits off extends to zero. Oh, then that&#x27;s the access. Where? Well, sign square. Dr X Over. It&#x27;s just weird. Ah, which can beat it. And as then three x were over. Its is where us? So it we&#x27;re five. It&#x27;s over. Excess were which can be also written as three times sign. It&#x27;s where the XT Where over three Xs. Where times one over or sign. But he excess were plus 25 Sign where or five x over 25. It&#x27;s extra square. Ah, the limit extends to therefore signed Square three. Answer. Spread over three. X squared is equal to one, and the limits were went over. Society X squared when extends to zero, is equal to one plus 25 times the limit for science. Weird five x over. Ah 25. It&#x27;s where 25 in its worst and beating. And that&#x27;s fine. X Would where so science were five X over five x. Older square is equal to one square, so the thunder answer would get three times 25 6 20 Hey,

Yaw Asomani · Answer

the limit as X goes to zero of sign sign three X sign five X over X squared. Right. So how do you go about this one? What they&#x27;re gonna use if you know, watched that last you tour before this one. Please watch. It was even. Do what? Exactly what we just did, right? We&#x27;re just gonna multiply this one by three x times five x over three x times five X Right. Once you do that, then we&#x27;re gonna have is sign three eggs over three X. Right, then times limit. Because you can separate it right. Then you&#x27;re gonna be a sign five x five x, then times what is left is limit. X goes to zero. What is left is, uh, this numerator here because we have distributed each of the denominator to, uh, these ones. Right. So what is that? Is this numerator here? And then this denominator here, that is what is left. So that is going to come here. Three x five x or X squared. You know, this is one we know this this one. So what is gonna be left is, um, limits Extra zero. I have three x times five X That is gonna be 15 15 x squared on the X squared. So this is canceling this. So finally the limit as X goes to zero, it&#x27;s just 15. Okay, She was 15.

R M · Answer

problem. Number 38 Limit extends to zero or it squared minus three. Sign X over X. Which community and as it&#x27;s squared over X minus Great sine X over x jay&#x27;s X minus three sine x over its the limit or extends to zero vaccine is the question. Zero linemen three times the limit off sine X over Axity. Put the one partner answer. Forget Bean.

Justin Leonesio · Answer

all right for this example we&#x27;re going to attempt to solve the limit is X approaches zero for this expression. And if you look at using the substitution method initially here, you&#x27;ll be able to tell pretty easily that you&#x27;ll get a zero on top and a zero on the bottoms. With indeterminate form like that, we&#x27;re gonna need to use something different. And in particular, they give us some tools here to be able to use it from the section and therm to rather that they give us. This limit was involved. This is a tool that they&#x27;re providing for us. And in previous examples, we&#x27;ve looked at creating kind of a brand new tool from this old one here. Notice that if sign of X over X is approaching one as the X values approach zero, well, then the reciprocal should also be true. It was X approaches. Zero. You flip this upside down. You should also be approaching one, because that means that roughly the top of the in the bottom are approaching the same result, leading us towards one overall. So this should hold to, and this both of these tools can be utilized within the problem if we need him. So to start actually working on this, it might be beneficial if we break it up into multiple pieces all multiplied together. So we&#x27;ll start with the limit as X approaches. Zero still. But we&#x27;re going to start kind of forming some of these, if at all possible. Now, a good way to start off here is Let&#x27;s just keep this sign of three X here as the numerator of one fraction. And to help out that expression little bit for the denominator, we&#x27;re gonna need a three x from the bottom, right? For in order for this to work out, we need the expression here within the sign also appear in the denominator, or vice versa. So we&#x27;re gonna need a three ex eventually over here underneath the sign of three eggs. So let&#x27;s just take this X that we already have and join it up on this side here. Always get a head start on that. Okay. Now, for the other parts noticed, we have really just this sign of five x on the bottom and the sign of two X on the top. Let&#x27;s start separating those into their own fractions. I&#x27;ll put the sign of five X here in the denominator for this first fraction. And then for the 2nd 1 we&#x27;ll put the sign of two eggs over here. Now, each of these pieces is gonna have something else that we&#x27;re gonna create in their denominator or numerator, respectively. So I&#x27;m just gonna leave them blank for now. Picture them is ones for the moment, if you like. Okay. Well, remember, we have just a X over here. We&#x27;re gonna need a three X down there in order to make this work. So what I&#x27;m gonna do is multiply the denominator by three. Acted by three, rather to create the three X. Don&#x27;t forget. If you do it there, you&#x27;re gonna have to also do it to the numerator at some point. So I&#x27;m gonna put it out back just to remind us that we did multiplied by three at some point. Okay? Now, if you take the limit is X approaches zero. This entire expression three X, both here and here will approach 02 So it fits this description here that we have and this whole portion will approach. One said this part here will approach one. We&#x27;re taking a limited just this portion on Lee as we learned that we could do in section 2.3 with some of those limit laws, particularly the product law. Okay. Now, for this one. Well, we would like to have five x on the numerator as well. So let&#x27;s go ahead and create that. If we put a five x up here, don&#x27;t forget, we&#x27;re gonna have to do it to the denominator as well, right? Multiplied on top and bottom. So I&#x27;ll keep the three here on top of thing. But this five x right below it. Okay, so now we&#x27;ve taken care everything. Everything is balanced out. And then as X approaches here, the entire five X will approach zero. So this will fit perfectly now, and so will allow this entire expression. It&#x27;ll approach one. Okay, well, we still have a little bit left over here. We&#x27;ll need a to X here in the denominator of this in order for toe workout. Now. Truthfully, if you want to, you can just put it to here and just grab this X from over here because that that will form a to X here, in the denominator of this that you could do that if you want to. Just for the sake of keeping everything nice and kind of pretty and together here, I&#x27;m just gonna put a next here along with the two. It&#x27;s not a problem, because I&#x27;ll just have to do it up here in the numerator. Not a big deal. We multiplied by two extra in the nominators. Same thing here. And now that we have this expression ready to go, we can utilize this tool and go ahead and let this approach positive on this well, and I will have to do really to finish it Averages, too. A little bit of canceling for these exes that we already have so x here. And the denominator can cancel with the sex in the numerator leaving one&#x27;s behind. And that means our numerator overall to be six and the denominator will be five. So that means our overall expression created this as our limit values six over five

Matt Just · Answer

Okay. Here have I wanted his ex of purchase Surprise zero The X minus X co sign eggs of here sine squared three x. Okay. I mean, I guess we should notice first that we try to plug in zero. We&#x27;re going to get zero over zero. No. So let&#x27;s see what we can do here. Well, the first thing, maybe to try is too down is probably too. Break it up. Okay. What? I mean by that I mean, uh, write this as the limit. It&#x27;s expert zero uh, x over sine squared three x. Okay. And then subtract the limit is experts zero of eggs cause I necks over sine squared. Re x. Okay, So what can we do here? Well, now this is signs squared. OK? So if I want to be able to I kind of do my usual trick and divine this by X. I actually need to Oh, divide by X squared, actually. Divide by three expert or nine x squared. So this is the limit. It&#x27;s expertise. Zero x over some of the multiply by nine x squared over nine x squared. Okay. And then minus the limit. The legs coast hero of X because the next time and did the same thing over here with nine x squared sine squared three X over nine x squared and let me make a note here said nine. Ah, I just thought, Why am I doing this? It&#x27;s a sign squared three X over nine X squared is the same thing as sign three X over three X squared, but this is going to want so therefore, this is going to want is one&#x27;s greatest one. So what we have Okay, so we have when? Over here. See, this is the limit. So what&#x27;s what&#x27;s left after I do this? So X over nine x squared and then minus pat. Okay, coz I nexus going to one That&#x27;s not a problem. And when x over nine x squared And now Okay, so I&#x27;m re combining the limits because actually, the linens excellent. Excellent x squared is going to infinity and then the same thing over here. So I have another indeterminate form, so I need to put him back together, and then these air actually going to zero right before I take the limit to the turtle in it is just zero

Issa Dababneh · Answer

So in this question, were asked to determine that limit as X goes to zero off X minus x co sign X divided by science squared three X And we&#x27;re also told that we should use the relationship right here. The limit right here that limit escape of zero sank a divided by key on. One immediate observation we can make is that the limited scale goes to C zero, their reciprocal. But what&#x27;s on the inside here is mostly people toe. This is found using the walls of the recipient the division. All right, so now the first thing we&#x27;re gonna do is we&#x27;re gonna try toe to break. We manipulate what we have on the inside. So, for example, we allowed an X and then we can also split sine squared of three Xs signed three extent sine three x. All right, now we can do some further manipulations. For example, we split them into three fractions and then now we&#x27;re gonna try to use the limit that we know just the monitors cables to zero sign K divided bike equals to one and the reciprocal that we&#x27;re going to try to use those so we can determine the limit. So we see here Swe have sine three x and the denominator and we only have X in the numerator. What we can do is multiplied by three divided by three so that we can get a three X and the numerator again. This is this is this is this is valid because we&#x27;re multiplying by three divided by 33 divided by three is just one So we&#x27;re simply multiplying by one which is okay we&#x27;re not changing anything. The same thing here we have a one in the numerator but we want we want it to be a three x So what we do is we multiply by three x divided by three x All right, and then now what we get It&#x27;s 1/3 times three x divided by sine three x And then one older again 1/3 times. Ah, so yes, so 1/3 times three x divided by side of three X And then and then what we have. So we took one accidents here and then we moved it right here. So it&#x27;s one minus co sign X divided by X So why did we do that? So the reason why is that? Now we plug everything back in. We know so one over threes accosted, so nothing changes. Now the living is exposed to 03 x, divided by sine of three acts. Well, this is just one, and then 1/3 again is a cause. That&#x27;s where the limit really doesn&#x27;t affect it. And then one more time. The limit is exposed to 03 x divided by sine of three x. This is again one, but what about one minus co sign X, Divided by X. What&#x27;s the limit of that is exposed to zero? Well, just a quick review if you you do. If you want to find the limit is exposed to 01 minus coastline X, divided by X. What you do is you multiply it by a convoluted or a sneaky work so you&#x27;re won&#x27;t you multiply it by one plus co sign X divided by one What&#x27;s coast? Now the numerator becomes one minus co sign square x Your foil, the numerator you get one minus coast side square X, which is just science and the denominator is X times one plus co science. And then now you try to do some further manipulations. So you already have sine X divided by X. This is fine. You only have Sinus. So what you do is you want X in the denominator so you multiply by X divided by X. And this is valid because X divide by X is just one. So now you have the limit of as X close to zero off sine X, divided by excesses one. So I next divided by excesses one. And now, if you plug in, zero for accident. Zero divided by one plus co Sign of zero. No sign of zeros Once you have one times one time zero divided by two. This is just zero. So now we know what&#x27;s the value of one minus co sign X Divided by X. This is just syrup. Yeah, 1/3 time&#x27;s one times 1/3 times one time zero I limit this simply zero

Problem

(a) Complete the table and make a guess about the…

Problem 40 Hard Difficulty

Find the limits.
$$
\lim _{x \rightarrow 0} \frac{\tan 3 x^{2}+\sin ^{2} 5 x}{x^{2}}
$$

Answer

28

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Calculus Early Transcendentals

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Precalculus Review - Intro

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Howard Anton, Irl Bivens, Stephen DavisCalculus Early Transcendentals

Grace H.

Anna Marie V.

Caleb E.

Samuel H.

Howard Anton, Irl Bivens, Stephen Davis

Calculus Early Transcendentals