evaluate-the-indicated-limit-if-it-exists-assume-that-lim-_x-rightarrow-0-fracsin-xx1-lim-_x-righ-15

Question

Evaluate the indicated limit, if it exists. Assume that $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$
$$\lim _{x \rightarrow 0} \frac{2 x}{3-\sqrt{x+9}}$$

Amy Jiang · Accepted Answer

Okay, so I supported the limit. Other following a precision. So if you use textile that get in the denominator three money, Cyril, her start three months, three, Which is a quick deal. So we get the vision myself. So instead of doing that, why don&#x27;t we multiply and divide by our contact base? And that&#x27;s three plus the square root of expose line. Okay, so that gives us, um let&#x27;s see Vore numerator as is. And that&#x27;s multiply out our denominator. Okay, so that gives us three times three, which is nine minus. Um, Marnix nine. Do we see our nine councils? And once or nine cancels, we can cancel out this, um, this X as well. So we get the limits as X approaches zero of 92 times three plus the square root of x plus. Fine. Okay. And I were using drugs off. We get negative, too. Times three plus The square root of joe coastline. That&#x27;s just nine. Some get negative two times. Did it for three. Which six, and that gives us negative 12

Yaw Asomani · Answer

so is getting more complicated as we go forward. But it is not. Anything difficult is toe requires a little bit of smartness from you, and you can you can sail through carefully and peacefully. Right now. This one is not a rational function. It is just a list, a linear function, right. Rational function is a function that you can have a photo like. Experts two or X minus three. Just irrational function. But it&#x27;s when there&#x27;s nothing over right. It&#x27;s just a linear function or they see whole function, right? So how do we do something like that? Well, we can still force away through they making it rational function if we decided to use congregate certain. So this is he just multiply it by the congregate. So once you do that, then you make it rational. Now, now, when you multiply the top one, you know that is gonna be the first term square. The first time squared is just that&#x27;s what records square root. So when you square, it is just gonna be the number of yourself, then minus the second term square. This is the second term. So when you square, it is gonna be nine x squared, right? And then over that right. You have this one and they taking the limit as X approaches infinity off such a thing. Right now we know that this is night positive. Nine x creators is negative. Nine x were so they&#x27;re gonna subtract out. So what is gonna go left here is just x here, right? And this is gonna be a nine x squared plus X I m plus three x and taking the limit as X goes to infinity right now, what is the highest power of X? The highest power of X is not X squared because of the square root. Right, So there&#x27;s power of aces this X you right? So the highest pile of eggs X there, then when you can do is divide each and every side but the numerator and the denominator by X. Right. So there&#x27;s gonna be X over eggs, and then the denominator is also gonna be That&#x27;s very tough. Nine X squared plus X right in close, three explosive over X. Like that. Right? So then this one is gonna be one, right, cause the top is gonna cancel each other. And in here we can we can share, right? We&#x27;re gonna share. So we share that this and then this is also gonna be so this is gonna take eggs, and this is gonna take X. We just share it amongst theist enumerators, and this cancels this one. So our problem here is this. Well, who was the last tutorial? You know how to kit this X inside the square. Right? So I&#x27;m just gonna do it straight away, So there&#x27;s gonna be limit, as except which is that it&#x27;s gonna meet that first is gonna be, uh, this do this. The new mayor is gonna be one. Right? So that is what you have that one for. And then here, this is gonna be a nine x squared plus eggs. We&#x27;re gonna push this exterior into the X squared by squaring it right? Remember, we do that in the last tutorial. So this is gonna be, uh, this is gonna be here. This is gonna be, uh, exclude as well. Right? And then close. My three left, Miss my three here. Right. So when I also share this X, where to each of these two terms, then I&#x27;m gonna have one over square root off nine X squared over X squared, then plus X over X squared, then plus three. So finding them having limits as X approaches Infinity one over. Swim route off. Now, this was gonna cancel this way, so I&#x27;m gonna have nine. Then this one is gonna cancel one of these is gonna be one over X, then plus three. So when I send X to infinity, then you can see that this one is gonna eventually go to zero. Right? So what I&#x27;m gonna have is one over a squared off nine plus three. What is good if I screwed of nine is to me. And three plus three is six services, 1/6.

Amy Jiang · Answer

find a limit of the phone. Call it. Oh, yeah, Rose here that you have minus and plus size. So why don&#x27;t we use are some and are different ball? Does he convinced interview off this limit? Okay. And now you see that we have a constant here, So why don&#x27;t we your, um our law of constant multiples about everything put out. And over here we can use our current law, which means that we could take our limits inside our power. We have the limits as expertise. Ural of extra part two on in one of three. It&#x27;s excellent. Zero of X and limits, like 40 of what? Okay, I know we can evaluate this. The limit as active with zero. We just place that. And so that&#x27;s still squared. 123 I&#x27;m Darryl. And then plus the limited constant. That&#x27;s that&#x27;s equal to our constant. That&#x27;s one. We have a general monitor all plus one for a limited with one

Subhadeepta Sahoo · Answer

in this course in order to find the limit actions to infinity we&#x27;re told off nine X squared minus X three X question Bressman Logic to this is conducted Muslim. My next clear minus X minus three x might depend by explosion. He looked at 19 Expect thanks. Last week&#x27;s they began by that same contribute mine ethics Just weeks. No, no. I should consider the memory of the bomb. It Let&#x27;s begin. Prime Minister. Yes. When was this quick? When you get my next week, many thanks My next night next week. Do you like good Nynex wit? Blessed be X So this town&#x27;s got canceled. Get Macduff Johnston, 30 minus x like, Got off, Nynex quipped, minus six. Blessed be X not dividing numerator and the number of my ex cream attacked after Internet. My next one join dryness when their ex looks. No one expects maturity. This Duncan 20 and I left with my next one Lake, who name Let&#x27;s can you go to minus 1 May six

Amy Jiang · Answer

Okay, so we&#x27;re asked to find the limits of the following equation. So it was here. If we use Drexel, we have division by zero. So instead, let&#x27;s see if you contract the routes. Are, um, following polynomial owes we have in our numerator and denominator. So notice here that we have X cubed minus one. That&#x27;s one to depart three. And then we can use the difference of two cubes. That is, we have X minus one times x squared plus X plus one. And we also need the limits of your out about that. Forget about that. That&#x27;s super unfortunates. And then we can also factor out our denominator into, um X minus or actually plus three and then x minus one. I know we can see that. In fact, there, out a X minus one. I know that X cannot be equal to one. No, um, we can use drugs off. So we got one squared plus one plus one over one plus three, and that gives us 3/4. So we get the following

Amy Jiang · Answer

we want to find limit of No, it&#x27;s here that we have a Cuban. So this nickel to power, why don&#x27;t we use our power law and bring in our limit vanishes by power. Okay, I think we have the limits as actually two of two x one. And now we noticed we have a somewhat between two terms that we can use our some ball that gives us you know, of limits as X approaches two of us, plus the limits as X approaches to of one noticed here that we have a constant. So why don&#x27;t we use our conference law here? We can bring that outside of our limits. That&#x27;s to limit as X approaches to off and then we&#x27;ll limit as X approaches to of one. What is that, People? The minutes as x two or X equal to two times to what? The limits of a constant that&#x27;s this equal to a constant it. Well, that&#x27;s one Yes, four plus one cube of five

Problem

Evaluate the indicated limit, if it exists. Assum…

Problem 14 Medium Difficulty

Evaluate the indicated limit, if it exists. Assume that $\lim _{x \rightarrow 0} \frac{\sin x}{x}=1$
$$\lim _{x \rightarrow 0} \frac{2 x}{3-\sqrt{x+9}}$$

Answer

-12

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Robert T. Smith, Roland B. Minton

Calculus: Early Transcendental Functions

Grace H.

Anna Marie V.

Caleb E.

Joseph L.

Precalculus Review - Intro

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Calculus: Early Transcendental Functions

Grace H.

Anna Marie V.

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Precalculus Review - Intro

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Robert T. Smith, Roland B. MintonCalculus: Early Transcendental Functions

Grace H.

Anna Marie V.

Caleb E.

Joseph L.

Robert T. Smith, Roland B. Minton

Calculus: Early Transcendental Functions