# One two-variable, and one three-variable limits

1. Mar 29, 2009

### libelec

1. I need to find these two limits, without using the definition of limit:

a) $$\mathop {\lim }\limits_{(x,y) \to (0,2)} \frac{{y^2 (x - 2)^2 }} {{x^2 (y - 2)^2 }}$$

b) $$\mathop {\lim }\limits_{(x,y,z) \to (0,1,1)} \frac{{y + 1}} {{\sqrt {z^2 - 1} }}$$
3. The attempt at a solution

For a) I found that if looking for the limit using curves, in this case using y-2=mx, with m=constant, and y-2=kx^2, with k=constant, are infinite.

For b) I'm clueless.

Any help will be thanked.

2. Mar 30, 2009

### CompuChip

For the second one, note that there is no x-dependency. So you can look at curves in the (y, z)-plane and give them arbitrary x-dependency, i.e. only consider
$$\lim_{(y, z) \to (1, 1)} \frac{y + 2}{\sqrt{z^2 - 1}}$$

3. Mar 30, 2009

### libelec

Yes, I was thinking about that, thanks.