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Chapter 12: Q. 13 (page 989)

Find the set of points where the function is continuous.
$f (x, y) = \frac{x + y}{x - y}$

Short Answer

Expert verified

$f (x, y) = \frac{x + y}{x - y} is continuous for all (x, y) in ℝ^{2} except when x = y .$

Step by step solution

01

Step 1. Given information is:

$f (x, y) = \frac{x + y}{x - y}$

02

Step 2. Finding Points

$Given function is rational function with denominator x - y . So f (x, y) is continuous at all points (x, y) in ℝ^{2} except at those points where x - y = 0 Since, x - y = 0 if x = y Therefore, f (x, y) = \frac{x + y}{x - y} is continuous for all (x, y) in ℝ^{2} except when x = y .$

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Most popular questions from this chapter

Find the gradient of the given function, and find the direction in which the function increases most rapidly at the specified point P.

$f (x, y) = \frac{y}{x}, P = (4, 3)$

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

$\lim_{(x, y, z) \to (0, 0, 0)} \frac{x^{2} + y^{2}}{x^{2} + y^{2} + z^{2}}$

Find the directional derivative of the given function at the specified point P in the direction of the given vector. Note: The given vectors may not be unit vectors.

$f (x, y, z) = \sqrt{\frac{x y}{z}}, P = (2, 3, 1), v = ⟨ 2, 1, - 2 ⟩$

Find the gradient of the given function, and find the direction in which the function increases most rapidly at the specified point P.

$f (x, y) = \frac{x}{y^{'}} P = (4, 3)$

Sketch the level curves f(x, y) = c of the following functions for c = −3, −2, −1, 0, 1, 2, and 3:

$f (x, y) = y^{2} - x^{2}$

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