find the limit, if it exists, or type dne if it does not exist.\na. $lim_{(x,y)\to(0,0)}\frac{(x +…

find the limit, if it exists, or type dne if it does not exist.\na. $lim_{(x,y)\to(0,0)}\frac{(x + 24y)^2}{x^{2}+576y^{2}}$\nb. $lim_{(x,y)\to(0,0)}\frac{9x^{3}+8y^{3}}{x^{2}+y^{2}}$\n(hint for part b: let $x = rcos\theta$, $y = rsin\theta$, and note that $x^{2}+y^{2}=r^{2}$).

find the limit, if it exists, or type dne if it does not exist.\na. $lim_{(x,y)\to(0,0)}\frac{(x + 24y)^2}{x^{2}+576y^{2}}$\nb. $lim_{(x,y)\to(0,0)}\frac{9x^{3}+8y^{3}}{x^{2}+y^{2}}$\n(hint for part b: let $x = rcos\theta$, $y = rsin\theta$, and note that $x^{2}+y^{2}=r^{2}$).

Answer

Explanation:

Step1: Convert to polar coordinates for part A

Let (x = r\cos\theta) and (y=r\sin\theta). Then (x^{2}+y^{2}=r^{2}) and ((x + 24y)^{2}=(r\cos\theta+24r\sin\theta)^{2}=r^{2}(\cos\theta + 24\sin\theta)^{2}). The limit (\lim_{(x,y)\to(0,0)}\frac{(x + 24y)^{2}}{x^{2}+576y^{2}}) becomes (\lim_{r\to0}\frac{r^{2}(\cos\theta + 24\sin\theta)^{2}}{r^{2}\cos^{2}\theta+576r^{2}\sin^{2}\theta}=\lim_{r\to0}\frac{(\cos\theta + 24\sin\theta)^{2}}{\cos^{2}\theta+576\sin^{2}\theta}). Since this limit does not depend on (r), we can find its value. Expand the numerator: ((\cos\theta + 24\sin\theta)^{2}=\cos^{2}\theta + 48\sin\theta\cos\theta+576\sin^{2}\theta). So the limit is (\frac{\cos^{2}\theta + 48\sin\theta\cos\theta+576\sin^{2}\theta}{\cos^{2}\theta+576\sin^{2}\theta}). If (\theta = 0), the limit is (\frac{1 + 0+0}{1+0}=1). If (\theta=\frac{\pi}{2}), the limit is (\frac{0 + 0+576}{0 + 576}=1). So (\lim_{(x,y)\to(0,0)}\frac{(x + 24y)^{2}}{x^{2}+576y^{2}} = 1).

Step2: Convert to polar coordinates for part B

Let (x = r\cos\theta) and (y = r\sin\theta). Then (x^{2}+y^{2}=r^{2}), (9x^{3}+8y^{3}=9r^{3}\cos^{3}\theta+8r^{3}\sin^{3}\theta). The limit (\lim_{(x,y)\to(0,0)}\frac{9x^{3}+8y^{3}}{x^{2}+y^{2}}) becomes (\lim_{r\to0}\frac{9r^{3}\cos^{3}\theta+8r^{3}\sin^{3}\theta}{r^{2}}=\lim_{r\to0}r(9\cos^{3}\theta + 8\sin^{3}\theta)). As (r\to0), regardless of the value of (\theta), (r(9\cos^{3}\theta + 8\sin^{3}\theta)\to0). So (\lim_{(x,y)\to(0,0)}\frac{9x^{3}+8y^{3}}{x^{2}+y^{2}}=0).

Answer:

A. 1 B. 0