Question

Help!! Find the limit, if it exists or explain the reason why it doesn't exist.

Answer 1

\[\lim_{(x,y) \rightarrow (1,1)}\frac{ xy-1 }{ 1+xy }\]

Answer 2

plug in

Answer 3

Ah, so the question is, is this function continuous in the neighborhood of (1,1)?

Answer 4

x->1 and y->1 since f(x,y) is continuous at (1,1) then you just plug in

Answer 5

Really? I just plug in x=1 and y=1?

Answer 6

Yes, since the function is defined around (1,1) there is no reason to not. In the previous problem, the function was undefined at (0,0)

Answer 7

I got 0/2, so is the limit zero?

Answer 8

yep

Answer 9

Wow I feel stupid

Answer 10

Thanks

Answer 11

you can do as long as the f(x,y) is continuous at (x=a,y=b)

Answer 12

http://tutorial.math.lamar.edu/Classes/CalcIII/Limits.aspx

I really think this site is also useful for this topic if you want to give it a read. I love paul's notes.

Answer 13

Can you help me with one more?

Answer 14

ok

Answer 15

Use polar coordinates to find the limit if it exists
\[\lim_{(x,y) \rightarrow (0,0)} (x^2+y^2)*\ln(x^2+y^2)\]

Answer 16

Can you read it? my equation button isn't working

Answer 17

So you know \[r^2=x^2+y^2\]
and \[x=rcos(\theta)\]
and \[y=rsin(\theta)\]

Answer 18

so you can replace the x^2+y^2 with r^2

if (x,y)->(0,0) what do you think (r,theta)->(?,?)

Answer 19

(0,0)?

Answer 20

well r is definitely 0

Answer 21

so i know we have (x,y)->(0,theta)
and that is really all we need for this problem

Answer 22

since your f(x,y) can be turned into f(r)

Answer 23

i'm having problems on this computer
i will return