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Find the following limit state that it does not existlim h_0Simplify the given limit9 + h lim h-0Simplify your answer)h-0...

Question

Find the following limit state that it does not existlim h_0Simplify the given limit9 + h lim h-0Simplify your answer)h-0

Find the following limit state that it does not exist lim h_0 Simplify the given limit 9 + h lim h-0 Simplify your answer) h-0



Answers

Evaluate the limit, if it exists.

$ \displaystyle \lim_{h \to 0}\frac{\sqrt{9 + h}-3}{h} $

Okay, so this is limits H 20 Uh, so you have nine. You want to do this one? It's if we want to do this one. What we gotta do here? This is a surge, right? Just assert, uh, turn. So we just have to do conjugation, right? Congregate served. So all we gotta do here, it's just multiply by to congregate, right? Because you get means that went right. See, close here. I'm gonna put mine is whenever i c minus. I'm gonna put plus, So I c minus here. I put plus here, Right? And then I divide by that same thing. This is the congregation I'm talking about. If you have. If you're familiar, research does how we do it, right? So once you have this one, then you just have to most apply rights. And when you do the multiplication, what is gonna happen? Well multiplied it the top and the bottom. It is just the first term squared. Mine is the second term square, right? Remember, you have a squared minus bees. Where is the same us? A minus? Bi a plus me as exactly what you're seeing here. Justice A minus B and then a plus B so you can just shrink it into this form. You have to be deft. Have to be smarter. You have to be sharper. Right? So that is what you have. So once you have that, then, um, what is happening at the denominator denominator here is gonna be a and h the night h plus three. Right? So what is gonna happen here is nine plus a try. When you square the square root, he's gonna have, uh, the number under a square side in the radical side and is three square is nine and gonna have h screaming of nine plus age in the street. So I have the limits off those to worry about, right, Nimitz age approaches zero off, You know, negative nine and negative night goes away cause nine minus 90 0 So I'm just gonna have h over this. Break eso age over nine plus h Inch of age. Times nine plus screaming Franklin H plus three here. Right. And this ages has a cancer in this one, right? So I just have finally limits age approaches zero off one over a scrambled off ni plus h plus three. Now, when I put Age equals zero. Although I have here is screwed off night. What is screwed off nine is three and then three plus three. Here is six. So then finally I have 1/6 us. My answer.

So we need to find the limit as H approaches zero of the quantity nine plus age. The square root -3 over age. So if we plug in zero directly, You will get the indeterminant form 0/0. So the way we are going to handle this is we are going to multiply mm square root of nine plus H plus three Over the screwed of nine plus each. That's three ceremonies planned by one. No big deal here. This comes to the nine plus H is will square. So that leaves us with nine plus H And -9 for the last terms. The middle terms will cancel the denominator is it's times The squirt of nine plus age. That's three. Okay, so this becomes the night and the night will cancel the age and the age will cancel. Stage will cancel that H. And you're left with emphasis, You're left with one over the square root of nine plus each plus three. Now when you plug in H0 you do not get an indeterminant form, get one over the spirit of nine plus three which is equal you 1/6

So we need to find the limit as h approaches zero of the quantity three plus H squared minus nine over age. So if we plug in H0 we will get the indeterminant form. zero over zero. So we need to simplify. Do some kind of factoring. So let's expand the numerator. Normally we start with factoring since the numerator is already factored, let's start with you standing. So the numerator would expand to nine plus six. Age plus H squared the minus nine. The denominator is speech. These nines will cancel. So now we are going to factor at an age so this is a judge six for each. Now our ages will cancel. Now we no longer have an indeterminant form so we can just plug in each zero zero plus six equals six. And we get The limit is equal to six.

We want to evaluate the limit of this expression as H approaches zero. Now if we try to directly substitute zero in for H uh we are going to get the indeterminant form of a limit. We're going to get 0/0. Uh So let's let's see if that's really the case plugging in zero for H If H 02 plus zero is to two Cubed would be eight. Okay so let's just write it down real quick. Okay if we just substitute zero in for age we would get two plus zero, cute minus eight over Once again plugging in zero for age. So this would be zero. Well two plus zero is too 2 to the 3rd is eight. Eight subtract 8 0. So we would have zero up top over zero. So directly substituting in zero for H uh brings us to the indeterminate form of a limit. 0/0. So basically we didn't get anywhere. So in order to actually calculate this limit, we are going to have to expand the numerator. We're actually going to uh do two plus H. Two the third and which means two plus eight times itself. Three times we're going to actually expand this and see if that helps us find the limit now. Two plus H. Two. The third is going to be a plus 12 H plus six H squared plus H. Cute. So two plus h. two. The third is this expression right here, we still have to write down to -8 and that whole thing. The whole expression gets put over each H and the denominator and let me fix this. Just supposed to be an L. Right here. So let's clean it up just a little bit. Okay so the limit of this expression that's H approaches zero is equal to the limit of this expression as H approaches zero. Now we can do a little bit of simplifying. We have eight here. Subtract eight there so those aides will cancel. And so we really had the limit of 12 H plus six. Eight square plus H cube. All divided by H. As H approaches zero. Each of these terms has an H. Uh has an H. So we are going to factor out the greatest common factor which is H. So rewriting this limit after we factor out the greatest common factor of H. Out of the numerator. Uh Well the numerator can be rewritten As aged times 12 plus six H. Plus each to the second and that is all over. H. You can distribute the multiplication by this age. To confirm that this is equal to what we had 12 times H. Six H. Times ages 68 squared eight square times ages. H. Cute. So the limit of this expression equals the limit of this expression now equals the limit of this expression. Well we can cancel out these ages times and by H. In the uh numerator. And dividing by agent. The denominator. Let's cancel out those H. Is So now we just have to take the limit of this expression as a jew approaches zero and we can take the limit of this expression. Uh As H approaches zero simply by plugging into zero everywhere you ch so 12 plus six times +06 times H will approach the limit of six times zero as H approaches zero plus a squared zero square. So now we are actually able to sub directly substitute zero in for H into this expression. Well uh six times 000 square to zero. And so our limit is 12. So the limit of this expression as H approaches zero is equal to 12.


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