Question

multivariable calculus:

showing that a limit DOES exist

Answer 1

image included

Answer 2

k so i'm not quite sure how to approach these problems because the epsilon delta method is easier to do in single variable calc

Answer 3

i thought you were supposed to match the equations up so that you would have \[\sqrt{x^2 + y^2}\] in both equations

Answer 4

the mane difference in limits for function of 2 variables, comparing with 1 variable, is that you can aprouch the limit not only from the left or right, but actualy from any direction. In the definition says that the function gona have the limit at some point if it is always the same, no mater what direction of aprouch you taking

Answer 5

right well i understand that but we need to generate a formal proof that it does exist from all directions using epsilon delta, sandwich theorem etc.

Answer 6

the formal proof is what i'm having difficulty with

Answer 7

so you intrested to find some kind of dependents that is actualy the distance from the point in question sqrt(x2+y2)

Answer 8

so all directions are treated at once

Answer 9

ah i see so it isn't really the epsilon delta method from before, well not exactly anyway

Answer 10

it is, just the delta is now dependent on x and y together, or better say: delta is the distance to the point

Answer 11

what you whant to proove is that the epsilon will be smole if this distance will

Answer 12

ahh right because before delta was just the distance on one axis? Now, you would use the distance from delta at any point because you aren't limited to the single axis?

Answer 13

right

Answer 14

you would still have to prove it in the same manner though, right?

Answer 15

ya

Answer 16

so, you'd have to take the epsilon equation and shift everything around so that it looks just like the delta equation

Answer 17

yes,you could see it this way. But the mane point is that you have to prove that you function values will be close enough to the limit point, if you take the delta smole enough

Answer 18

hrm, i think i'm going to run to the math lab real fast. it's kind of difficult doing this over computer :/

Answer 19

thank you very much!

Answer 20

yw