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Chapter 12: Q 42. (page 931)

In Exercises 41鈥�46, use polar coordinates to analyze the given limits.
$\lim_{(x, y) 鈫� (0, 0)} \frac{y^{2}}{x^{2} + y^{2}}$

Short Answer

Expert verified

The $\lim_{(x, y) 鈫� (0, 0)} \frac{y^{2}}{x^{2} + y^{2}}$ does not exist.

Step by step solution

01

Step 1. Given inforantion.

We have given expression : $\lim_{(x, y) 鈫� (0, 0)} \frac{y^{2}}{x^{2} + y^{2}}$

02

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

The relation between the rectangular coordinates $(x, y)$ and the polar coordinates $(r, 胃)$ is

$x = r \cos 胃 y = r \sin 胃$

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

role="math" localid="1653208927302" $\lim_{(x, y) 鈫� (0, 0)} \frac{y^{2}}{x^{2} + y^{2}} = \lim_{r 鈫� 0} \frac{{(r \sin 胃)}^{2}}{{(r \cos 胃)}^{2} + {(r \sin 胃)}^{2}} = \lim_{r 鈫� 0} \frac{r^{2} \sin 胃^{2}}{r^{2} \cos^{2} 胃 + r^{2} \sin^{2} 胃} = \lim_{r 鈫� 0} \frac{r^{2} \sin^{2} 胃}{r^{2} (\cos^{2} 胃 + \sin^{2} 胃)} = \lim_{r 鈫� 0} \sin^{2} 胃 = \sin^{2} 胃$

Since the value of $\lim_{(x, y) 鈫� (0, 0)} \frac{y^{2}}{x^{2} + y^{2}}$ depends on $胃$ . Therefore the value of $\lim_{(x, y) 鈫� (0, 0)} \frac{y^{2}}{x^{2} + y^{2}}$ is different for different value of $胃$ .

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Most popular questions from this chapter

$A chain rule for functions of two variables : Let f (x, y) = x^{2} y^{3}, x = s \cos t, and y = s \sin t . Substitute the expressions for x and y into f (x, y) .$

Evaluate the following limits, or explain why the limit does not exist.

$\lim_{(x, y) 鈫� (0, 0)} \frac{x + y}{x^{2} + y^{2}}$

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$Recall that the volume, V, of a cylinder is given by V (r, h) = 蟺 r^{2} h . Find \frac{鈭� V}{鈭� r} and \frac{鈭� V}{鈭� h} . What are the physical interpretations of these partial derivatives ?$

Solve the exact differential equations in Exercises 63鈥�66. $e^{y} + (x e^{y} 鈭� 7) \frac{d y}{d x} = 0$

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