Multivariable Indefinite Limits

[train449]

Homework Statement

Evaluate the following limits or determine that the limit does not exist.

b) lim (x,y)–>(0,0) (1+xy)^(1/xy)

Homework Equations

The Attempt at a Solution

I have a funny feeling this limit might exist. (Then again I get that feeling about every indefinite limit...) Can this be simplified further?

I tried converting ƒ to e^log ƒ to get rid of the exponent. Work attached.

Is there a better way of dealing with 1^∞ indefinite limits in more than one variable? Something with more of a multivariable calc flavor?

 

[SammyS]

Doesn't this remind you of lim_x→0(1+x)^1/x ?

By the Way: If you do your last step correctly, you merely get the original problem back.

 

[train449]

It took me forever, but I remember know! e

I had learned e as lim x→∞ (1+1/x)^x so it didn't quite jump out.

i know the limit of F as (x,y) → (0,y) = y + e (x,y) → (x,0) = x + e

How can I combine these results?

Or is that making things unnecessarily complicated?

 

[SammyS]

On your attached work sheet:
The line with the following is correct.

$\displaystyle \lim_{(x,y)\to(0,0)\,}e^{\displaystyle \frac{1}{xy}\ln\left(1+xy\right)}$​

The next line is incorrect. It should be either

$\displaystyle \lim_{(x,y)\to(0,0)\,}\left(e^{\frac{1}{xy}}\right)^{\ln\left(1+xy\right)}$

or else what you started with.​

The point is to get

$\displaystyle e^{\left\{\displaystyle \lim_{(x,y)\to(0,0)\,}\left(\frac{1}{xy}\ln\left(1+xy\right)\right)\right\}}$​

I've tried working with this in several ways and always get e, but I'm still not convinced it is e.

 

[HallsofIvy]

Would you feel better if you let u= xy and looked at
$$\lim_{u\to 0} (1+ u)^{1/u}$$

 

[train449]

I would feel A LOT better. What sort of 'approach' is that substitution?

Also would I need to find another path to see if the limit truly exists. I know when you substitute r = x^2 + y^2 you approach from all sides but I'm not quite sure how to visualize u =xy

 

[HallsofIvy]

Well, I would call it "substitution"!

And "finding another path" would not tell you it converges. Finding two different paths where the limit, along that path, is different tells you there is NO limit. But it is possible, in two or more dimensions, to have a limit where, approaching the point along an infinite number of paths (even every possible straight line) gives the same result yet there exist at least one other path where you get a different result so there is no limit.

Here, no matter what path you take, as x and y both go to 0, xy goes to 0. This is really just a single variable limit!