evaluate-the-limit-lim-_x-rightarrow-0-fracsin-5-x-sin-2-xsin-3-x-sin-5-x

Question

Evaluate the limit.
$$
\lim _{x \rightarrow 0} \frac{\sin 5 x \sin 2 x}{\sin 3 x \sin 5 x}
$$

Fuzail Shakir · Accepted Answer

canceling the side of five x er. We need to apply here too. So if you multiply the entire limit but two divided by two ex as well as three X divided by three, we can end up with the limit as X approaches. Zero off this sign of two X divided by two ex multiplied by 2/3 multiplied by three x, divided by the sine of three x, so this is equal to one tends to third Time&#x27;s one, which is equal to 2 30

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

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.

Zachary Mitchell · Answer

So we have this limit as X goes to zero of signed two X divided by signed five X First, let&#x27;s check what we get. If we try to do the limit, we get zero in the top zero that about on the bottom. So we have 0/0. That&#x27;s an indeterminant form, and we&#x27;re good to go to use low petals rule. Now, since we&#x27;re using loopholes will miss next step. Let me just go ahead and write L h stands for low patella or the equal sign just showing that this next step uses open cultural So the dirt of the numerator is to co sign to x. The derivative of the Dominator is five co sign five x So this is a limit that we can evaluate. We get 2/5 and we&#x27;re done

Matt Just · Answer

Okay, Lim is X approaches here. A sign? Five x over sign. It works. Okay, so this is zero over zero. But just try to plug in zero. And what I want to do is use. Okay, so I want to be able to divide sign five x over five exit and a good one and then sign for X over for X. That also goto one. So this is just a massive rewriting the limit. Thanks for two zero of Well, I have five. Thanks. Time&#x27;s sign five x over by VIX over for X times Sign for X over for ex okay and change anything. But now I see that this is going to want and this is goingto one. So what am I left with? These access actually. Cancel. So I slept with the limit. Zac&#x27;s a virgin zero before, which is five over four

Issa Dababneh · Answer

So this question were asked to determine the limit at his ex Goes to zero off sign five X divided by science or X, and we&#x27;re supposed to use this favor still right here. So the live it is cables, zero scientific. Your body that keep was one. Well, what follows from that limit is that we can take the reciprocal. So the limit of the reciprocal so that Limited&#x27;s cake goes to zero K divided by sliding K basic was when this was immediately from both laws off limits. All right, so now we&#x27;re gonna take what&#x27;s on the inside of the living, and then we&#x27;re gonna try to algebraic. So what we can do is we can split the what&#x27;s on the inside of the limited to two parts. So sign five x divided by one times one divided by sine forex. And then So we want. So we have Sinus divided. Signed five x divided by one. We want somehow down five. Excellent. You know the leader. What we can do is we can divide by five extra, but we can multiply by five x divided by five x and then we have one divided by sine forex again What we can do We want a four x in the numerator. What we can do is multiplied by four X divided by forex Because remember five x divided by five x and four x divided by four x These are just one so you can multiply by one and it won&#x27;t change in. So now we have a five x ray here on the outside we can pull this barbeques and the denominator of the inside. So we have this regularly the same thing. We could leave the one divided by forex in the denominator. So for excellent denominator that pull for excellently inside. And then finally we were rearrange everything we have bind about it by four time side of five s If I could play five x times four x divided by side forex And then now what we do is we take the living of that. So let&#x27;s see the limit as X close to zero Side five X divided by finance. Well, this is just what four x divided by side of forex again, this is just one. And then we have a concept for here by divided by four. Remember the limit of a constant. It&#x27;s just a constant itself. So our final on the final solution as the limit is X close to zero box I in five x divided by sine forex just five of

Problem

Evaluate the limit. $$ \lim _{x \rightarrow 0} …

Problem 42 Easy Difficulty

Evaluate the limit.
$$
\lim _{x \rightarrow 0} \frac{\sin 5 x \sin 2 x}{\sin 3 x \sin 5 x}
$$

Answer

$\lim _{x \rightarrow 0} \frac{\sin 5 x \cdot \sin 2 x}{\sin 3 x \cdot \sin 5 x}=\frac{2}{3}$

Related Courses

Calculus for AP

Related Topics

Discussion

Top Calculus 1 / AB Educators

Grace H.

Catherine R.

Caleb E.

Samuel H.

Calculus 1 / AB Courses

Precalculus Review - Intro

Functions on the Real Line - Overview

Recommended Videos

Watch More Solved Questions in Chapter 2

Video Transcript

Jon Rogawski & Ray Cannon

Calculus for AP

Grace H.

Catherine R.

Caleb E.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

What do you need help with?

Textbooks

Ask a question

Courses

AI Tutor

$\lim _{x \rightarrow 0} \frac{\sin 5 x \cdot \sin 2 x}{\sin 3 x \cdot \sin 5 x}=\frac{2}{3}$

Calculus for AP

Grace H.

Catherine R.

Caleb E.

Samuel H.

Precalculus Review - Intro

Functions on the Real Line - Overview

Jon Rogawski & Ray CannonCalculus for AP

Grace H.

Catherine R.

Caleb E.

Samuel H.

Jon Rogawski & Ray Cannon

Calculus for AP