5

3) Sketch the level surface function f(x,y,2) =2-x2 _y? at c = f(,-1,0)x2 _yl . Find the follwing limits: lim ()+(3,4) Vx? +y2xly- <8) lim (K,)(,) x" +y2...

Question

3) Sketch the level surface function f(x,y,2) =2-x2 _y? at c = f(,-1,0)x2 _yl . Find the follwing limits: lim ()+(3,4) Vx? +y2xly- <8) lim (K,)(,) x" +y2

3) Sketch the level surface function f(x,y,2) =2-x2 _y? at c = f(,-1,0) x2 _yl . Find the follwing limits: lim ()+(3,4) Vx? +y2 xly- <8) lim (K,)(,) x" +y2



Answers

Sketch the level surface $f(x, y, z)=k$ $$ f(x, y, z)=z-x^{2}-y^{2}+4 ; k=7 $$

To schedule level surface of F one case equals zero you can that their people to zero I will have noticed here is that this looks similar to the equation from elliptical Cool. That equation would be X squared over a squared last. Why squared or be square minus Z squared over C squared and in this case, A, B and C would be equal to zero. So I am expecting that the level surface is indeed on elliptical Cool. So using the 30 water, I grabbed the level surface and as perspective, it isn't in the local

So we're asked to use the limits that were given to sketch a graph of the function after sex. First, we're given that the limit of f of X as X approaches one equals infinity. I'm going to mark this limit in red. So this means that the limit of FX as X approaches one from both the sides greater and less than one equals infinity. And we're to draw this. We can show that as Ex gets closer and closer to one from the left side, why gets closer and closer to insanity? And as Ex gets closer and closer to one from the left side, why also gets closer and closer two in city, we've draw that just to make the lines a bit smoother. All right. Now, for the second limit were given were given that the limit of FX as X approaches three from the left side is equal to zero as X approaches three from the left side. Why gets closer and closer to zero, but never actually equals zero. Therefore will draw a little open dot right where the zero would be to show that why never actually reaches zero. Finally, the limits of f of X as X approaches three from the positive side is equal to negative infinity. We'll mark this in blue In order to draw this, we can show that as Ex gets closer and closer to positive three why it gets closer and closer to negative infinity and that is a sketch of the graph.

The sketch in Doesn't service of F at Keiko's wound. Don't take our equation instead of equal to one, and you can sell Posey. So that is the equalled negative for plus two white. That's one. Okay, so all the powers are are one. So this would be a plane, and it will intersect the Z axis. Uh, 001 Using the three d butter, you can sketch the actual curve, and here we have a picture. The plane.

Expression of for the sketch this and then check and see if the limit exists. And so if we're doing that here, and you can see that this simplifies to X squared. We could take the limit as X approaches zero here. That does in fact equals zero. So what that means is that the sketch of this Will look like having a whole lot zero and it will have the rest of the function go on from there. And we see as we approach from the left, The limit as X approaches zero from the left, zero. And that's also equal to the limit As X approaches zero from the right. And so by definition here, we have that since the limit from the left equals limit from the right, then the limit does exist here, even though it doesn't have an exact value at zero here, it doesn't matter. The fact that it's approaching it from both sides is enough to show that the limit does exist right? And so it in fact Equals to zero that we had here. So that's our answer there.


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