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• Notes

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 $y \geq x^{2}, z>0$