Calculus CLimits And Continuity Problem

Q1: Find the limit if it exist? 

$$\lim _{(x, y) \rightarrow\left(2, -1\right)} \frac{x^{2} y+x y^{2}}{x^{2}-y^{2}} $$

$$\lim _{(x, y) \rightarrow(2, -1)} \frac{4 \times -1+2 \times 1}{4-1} $$

$$=\frac{-4+2}{3} $$

$$=\frac{-2}{3} $$

Q2: Find the limit if it exist?

$$\lim _{(x, y, z) \rightarrow(0, 0, 0)} \frac{x y+y z}{x^{2}+y^{2}+z^{2}} $$

(1) $$\lim _{(x, y, z) \rightarrow(0,0,0)} \frac{0+0}{0}=\frac{0}{0}$$

(2) check 

$$x=y$$
$$x=2$$

$$\lim _{\left(x, y, z\right) \rightarrow (0,0,0)} \frac{x^{2}+x^{2}}{x^{2}+x^{2}+x^{2}} $$

$$=\frac{2 x^{2}}{3 x^{2}}=\frac{2}{3} $$

$$x=0$$
$$y=0$$

$$\lim _{\left(x, y, z\right) \rightarrow (0,0,0)} \frac{0+0}{0+0+z^{2}}=0$$

Limit doesn't exist

Q3: Determine the set of points at which the function is continuous

$$f(x, y)=\frac{x y}{1+e^{x-y}} $$

Req: {points} $$\Rightarrow$$ continuous

(1) $$1+e^{x-y} \neq 0$$

(2) $$1+e^{x-y}>0, \quad e^{x-y}>0 \quad$$ for all $$(x, y)$$

(3) Continass on $$R^{2}$$

Q4: Determine the set of points at which the function is continuous

$$f(x, y, z)=\sqrt{y-x^{2}} \ln Z$$

(1)

1. Limil exist
2. $$f(x, y, z)$$ defined.
3- $$\lim f(x, y, z)=f(x, y, z)$$

(2) $$y-x^{2} \geq 0$$

$$y \geq x^{2} $$

(3) $$\ln z \rightarrow z>0$$

$$f(x, y, z)$$ continuous $$y \geq x^{2}, z>0$$