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Chapter 12: Q 46. (page 931)

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

Short Answer

Expert verified

The value of $\lim_{(x, y) 鈫� (0, 0)} \frac{\sin (x^{2} + y^{2})}{x^{2} + y^{2}}$ is $1$ .

Step by step solution

01

Step 1. Given information.

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

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.

$\lim_{(x, y) 鈫� (0, 0)} \frac{\sin (x^{2} + y^{2})}{x^{2} + y^{2}} = \lim_{r 鈫� 0} \frac{\sin (r^{2} \cos^{2} 胃 + r^{2} \sin^{2} 胃)}{r^{2} \cos^{2} 胃 + r^{2} \sin^{2} 胃} = \lim_{r 鈫� 0} \frac{\sin r^{2} (\cos^{2} 胃 + \sin^{2} 胃)}{r^{2} (\cos^{2} 胃 + \sin^{2} 胃)} = \lim_{r 鈫� 0} \frac{\sin r^{2}}{r^{2}} = \lim_{r 鈫� 0} \frac{2 r \cos (r^{2})}{2 r} = \lim_{r 鈫� 0} \cos (r^{2}) = \cos 0 = 1$

Since the value of $\lim_{(x, y) 鈫� (0, 0)} \frac{\sin (x^{2} + y^{2})}{x^{2} + y^{2}}$ is $1$ .

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Most popular questions from this chapter

Given a function of n variables, $z = f (x_{1}, x_{2}, 鈥�, x_{n}),$ and a constraint equation, $g (x_{1}, x_{2}, 鈥�, x_{n}) = 0,$ how many equations would we obtain if we tried to optimize f by the method of Lagrange multipliers?

How do you find the critical points of a function of two variables, $f (x, y)$ ? What is the significance of the critical points?

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