Q 44. - Calculus

Question

In Exercises 41–46, use polar coordinates to analyze the given limits.

$\underset{(x,y)\to \left(0,0\right)}{\lim } \frac{x^2{y}^3}{x^4+2x^2{y}^2+y^4}$

Step-by-Step Solution

Answer

The value of $\underset{(x,y)\to \left(0,0\right)}{\lim } \frac{x^2{y}^3}{x^4+2x^2{y}^2+y^4}$ is 0.

Step 1. Given information.

We have given expression :  $\underset{(x,y)\to \left(0,0\right)}{\lim } \frac{x^2{y}^3}{x^4+2x^2{y}^2+y^4}$

Step 2. Use polar coordinates to analyze the given limits.

 The relation between the rectangular coordinates $\left(x,y\right)$ and the polar coordinates $\left(r,\theta \right)$ is

$$\begin{gathered} x=r \cos \theta \\ y=r \sin \theta \end{gathered}$$

On substituting values of $x$ and $y$ we get.

$$\begin{gathered} \underset{(x,y)\to \left(0,0\right)}{\lim } \frac{x^2{y}^3}{x^4+2x^2{y}^2+y^4}=\underset{r\to 0}{\lim } \frac{r^2{\cos }^2\theta \left(r^3{\sin }^3\theta \right)}{r^4{\cos }^4\theta +2r^2{\cos }^2\theta \left(r^2{\sin }^2\theta \right)+r^4{\sin }^4\theta } \\ =\underset{r\to 0}{\lim } \frac{r^{5 }{\cos }^2\theta {\sin }^3\theta }{r^4({\cos }^4\theta +{\sin }^4\theta +2{\sin }^2\theta {\cos }^2\theta )} \\ =\underset{r\to 0}{\lim } \frac{r^{ }{\cos }^2\theta {\sin }^3\theta }{({\cos }^4\theta +{\sin }^4\theta +2{\sin }^2\theta {\cos }^2\theta )} \\ =0 \end{gathered}$$

Since the value of $\underset{(x,y)\to \left(0,0\right)}{\lim } \frac{x^2{y}^3}{x^4+2x^2{y}^2+y^4}$ is $0$.