Approaching Limits with Multiple Variables: How to Choose the Right Method?

[loxagos_snake]

OK, our Calc professor gave us a paper with some limits to calculate, but I'm not quite sure about them...Here are some examples:

a)lim (5x^5+3y^5)/(x^2y^2+1)
x->0
y->0

Since the obvious replacement is too suspicious, I thought about looking if it exists by using the y=mx line. Indeed, for different values of m, the limit remained 0, but still I'm not sure if this was the right way.

b)lim (5xy+yz)/(x^2+xz)
x->0
y->0
z->0

I have absolutely no clue about this, except it brings spherical coordinates to my head.

One question I wanted to ask is how I should know which method to use on each limit? Like lines,parabolas,polar coordinates etc.


Anyway, thanks in advance and I apologize if the question format is wrong, it's just my first post!

 

[Dick]

For the first one, the obvious is correct. However (x,y) approaches (0,0) the numerator goes to zero and the denominator goes to 1. It's not really an indefinite limit. For the second one, put z=0. Now think about different ways x and y can go to zero. There's not really a fixed 'method' for these problems. You just have to try to visualize how the function behaves by trying different things.