1. multivariable limit

not too sure how to do limits in multivariable yet.

heres my question?:

Limit as (x,y)-->(0,0) of (e^(xy)-1)/y.

I know the answer is zero from the back of the book but how i did it it got negative infinite.

I multiplied by x/x so the function was (xe^(xy)-x)/(xy) and split it into two limits. The first limit (xe^(xy))/xy goes to 0 i think, and then -x/(xy) i think goes to - infinity.

how do i actually do this/whats wrong with what i did?

2. yes, the limit is zero, but it's enough to remember a little fact from single variable calculus: $\dfrac{e^x-1}x\xrightarrow[x\to0]{}1$ so write $\displaystyle\frac{{{e}^{xy}}-1}{y}=x\cdot \frac{{{e}^{xy}}-1}{xy},$ and the rest follows.

3. oh! right! thank, that was much more simple than i was making it out to be

4. It is false that...

$\displaystyle \lim_{(x,y) \rightarrow (0,0)} \frac{x\ e^{x\ y}}{x\ y}=0$

The problem is easily solved in polar coordinates setting...

$x= \rho\ \cos \theta$

$y= \rho\ \sin \theta$

Kind regards

$\chi$ $\sigma$