find $lim_{(x,y)\to(0,0)}\frac{10x^{3}-10y^{3}}{x - y}$.
find $lim_{(x,y)\to(0,0)}\frac{10x^{3}-10y^{3}}{x - y}$.
Answer
Explanation:
Step1: Factor the numerator
We know that $a^{3}-b^{3}=(a - b)(a^{2}+ab + b^{2})$. So, $10x^{3}-10y^{3}=10(x^{3}-y^{3})=10(x - y)(x^{2}+xy + y^{2})$.
Step2: Simplify the function
Substitute the factored - form into the original limit: [ \begin{align*} \lim_{(x,y)\to(0,0)}\frac{10x^{3}-10y^{3}}{x - y}&=\lim_{(x,y)\to(0,0)}\frac{10(x - y)(x^{2}+xy + y^{2})}{x - y}\ &=\lim_{(x,y)\to(0,0)}10(x^{2}+xy + y^{2}) \end{align*} ]
Step3: Evaluate the limit
Substitute $x = 0$ and $y = 0$ into $10(x^{2}+xy + y^{2})$: [10(0^{2}+0\times0 + 0^{2})=0]
Answer:
$0$