Limit of function

Exercise

1.

Determine whether the following sets are open set, closed set, bounded set, connected set, or region. Then find the boundary and closure.

(a)

$${D = \{(x,y) : 0 < x^2 + y^2 < 1 \}}$$

(b)

$${D = \{(x,y) : x y \leq 0 \}}$$

2.

For $(x,y) \rightarrow (0,0)$, find the limit of the following functions.

(a)

$${\frac{\sqrt{xy}}{x^2 + y^2}}$$

(b)

$${\frac{xy}{x^2 + y^2 + y^4} }$$

(c)

$${\frac{xy}{x^2 + y^2 + y}}$$

3.

Determine the folowing functions are continuous at $(0,0)$.

(a)

$${f(x,y) = \left\{\begin{array}{cl} \frac{x^2y}{x^2+y^2}, & (x,y) \neq (0,0)\\ 0, & (x,y) = (0,0) \end{array}\right.}$$

(b)

$${\ f(x,y) = \left\{\begin{array}{cl} \frac{x^2-y^2}{x^2+y^2}, & (x,y) \neq (0,0)\\ 0, & (x,y) = (0,0) \end{array}\right.}$$

(c)

$${\ f(x,y) = \left\{\begin{array}{cl} xy \log(x^2 + y^2), & (x,y) \neq (0,0)\\ -1, & (x,y) = (0,0) \end{array}\right.}$$