Epsilon-delta proof for limits (multivariable)

  • #1
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0

Homework Statement

:
the question wants me to prove that the limit of f(x,y) as x approaches 1.3 and y approaches -1 is (3.3, 4.4, 0.3). f(x,y) is defined as (2y2+x, -2x+7, x+y).
[/B]

The attempt at a solution: This is the solution my lecturer has given. it's not very neat, sorry.
http://imgur.com/CfmIodw

So far, I've managed to understand what's going on, up till the part that says choose delta as min{1, epsilon/11} I'm really not sure how he got the 11 over there in the denominator. I can see how it fits at the last line of the page and it makes sense. but How did he get it firstly? Is it arbitrary, can it also be something like 10, 12 etc?

Sorry if this seems silly.
 

Answers and Replies

  • #2
34,673
10,809
##\displaystyle \frac \epsilon {12}## would work as well. 10 as denominator gives a value that is too large.

How to get 11: experience. If in doubt, take 1000. Or leave the denominator blank and fill it out once you got to the last line.
Such a prefactor is irrelevant - you'll also often see proofs that end with ##|a-b| < 11 \epsilon## instead of having the 11 earlier. That is perfectly fine.
 
  • #3
SammyS
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Homework Statement

:[/B]
the question wants me to prove that the limit of f(x,y) as x approaches 1.3 and y approaches -1 is (3.3, 4.4, 0.3).
f(x,y) is defined as (2y2+x, -2x+7, x+y).


The attempt at a solution: This is the solution my lecturer has given. it's not very neat, sorry.
http://imgur.com/CfmIodw

So far, I've managed to understand what's going on, up till the part that says choose delta as min{1, epsilon/11} I'm really not sure how he got the 11 over there in the denominator. I can see how it fits at the last line of the page and it makes sense. but How did he get it firstly? Is it arbitrary, can it also be something like 10, 12 etc?

Sorry if this seems silly.[/B]
Hello Zeeree. Welcome to PF !

I see that the function is given as
upload_2016-7-12_21-31-52.png


At the very least, you should use ^ to indicate exponentiation. Even better, use the X2 feature or LaTeX.

f(x,y) = (2y2+x, -2x+7, x+y) .
 

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