Limits in 2 variables.

[Overlord_Prime]

I have this sum: [LATEX] lim_{(x,y)->(0,0)} tan^{-1} (\frac{|x|+|y|}{x^2+y^2}) [/LATEX]

and here's my attempt at a solution using the limit definition:

I'll work with whats inside the arctan first, then dump the limit in there..I guess my limit as 0 (for fun--how to make intuitive guesses for proposed limits, I do not know). So

[LATEX] lim_{(x,y)->(0,0)}\frac{|x|+|y|}{x^2+y^2} [/LATEX]

[LATEX]|\frac{|x|+|y|}{x^2+y^2}|<\epsilon\ \ whenever\ \ \sqrt{x^2+y^2}<\delta [/LATEX]

[LATEX]\frac{|x|+|y|}{x^2+y^2}< \frac{\sqrt{x^2}+\sqrt{y^2}}{x^2+y^2}[/LATEX]

[LATEX]<\frac{\sqrt{x^2+y^2}+\sqrt{y^2+x^2}}{x^2+y^2}[/LATEX]

[LATEX]=\frac{2 \sqrt{x^2+y^2}}{x^2+y^2} = \frac{2}{\sqrt{x^2+y^2}}=\frac{2}{\delta}[/LATEX]

So, taking [LATEX] \delta=\frac{2}{\epsilon}[/LATEX]

The limit should be 0.

so my initial functions limit would be arctan(0)=0.

 

Now wolfram alpha tells me that my limit does not exist, and my textbook tells me that my limit exists and is equal to [LATEX]\frac{\pi}{2}[/LATEX]

What's going on ??!

[Xittenn]

When you plugged your question into wolfram you omitted the arctangent. The value inside leads to a certain value that could be called . . . . maybe what wolfram alpha called it, non-existent. Another way of saying it would be the limit as x and y approaches 0 of [math] \frac{|x|+|y|}{x^2+y^2} [/math] tends to . . . .

 

If you were to plug this value back into the original problem you would get a value not quite unlike . . . .

 

I am not immediately familiar with the method( or identities or anything, right now my brain is computing funny little images ) you have used to solve the problem. I have however, used elementary mathematics to deduce that in fact the answer in the back of the book is the correct one.

Edited December 1, 2011 by Xittenn