find the limit (enter dne if the limit does not exist) hint: rationalize the denominator. $lim_{(x,y) o(0,0)}\frac{-2x^{2}+2y^{2}}{sqrt{(-2x^{2}+2y^{2}+1) - 1}}$

Question

find the limit (enter dne if the limit does not exist) hint: rationalize the denominator. $lim_{(x,y)	o(0,0)}\frac{-2x^{2}+2y^{2}}{sqrt{(-2x^{2}+2y^{2}+1) - 1}}$

Accepted Answer

# Explanation:
## Step1: Rationalize the denominator
Multiply the numerator and denominator by the conjugate of the denominator $\sqrt{- 2x^{2}+2y^{2}+1}+1$.
$$
\begin{align*}
&\lim_{(x,y)\to(0,0)}\frac{-2x^{2}+2y^{2}}{\sqrt{-2x^{2}+2y^{2}+1}-1}\times\frac{\sqrt{-2x^{2}+2y^{2}+1}+1}{\sqrt{-2x^{2}+2y^{2}+1}+1}\
=&\lim_{(x,y)\to(0,0)}\frac{(-2x^{2}+2y^{2})(\sqrt{-2x^{2}+2y^{2}+1}+1)}{(-2x^{2}+2y^{2}+1)-1}\
=&\lim_{(x,y)\to(0,0)}\frac{(-2x^{2}+2y^{2})(\sqrt{-2x^{2}+2y^{2}+1}+1)}{-2x^{2}+2y^{2}}
\end{align*}
$$
## Step2: Simplify the expression
Cancel out the non - zero factor $-2x^{2}+2y^{2}$ (for $(x,y)
eq(0,0)$).
$$
\begin{align*}
&\lim_{(x,y)\to(0,0)}(\sqrt{-2x^{2}+2y^{2}+1}+1)
\end{align*}
$$
## Step3: Evaluate the limit
Substitute $x = 0$ and $y = 0$ into the simplified expression.
$$
\begin{align*}
&\sqrt{-2(0)^{2}+2(0)^{2}+1}+1=\sqrt{1}+1=2
\end{align*}
$$
# Answer:
$2$

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QUESTION IMAGE

Question

Explanation:

Step1: Rationalize the denominator

Step2: Simplify the expression

Step3: Evaluate the limit

Answer: