4.2 Limits and continuity (Page 5/14)

Calculus volume 3 Page 5 / 14

Show that the functions $f (x, y) = 2 x^{2} y^{3} + 3$ and $g (x, y) = {(2 x^{2} y^{3} + 3)}^{4}$ are continuous everywhere.

The polynomials $g (x) = 2 x^{2}$ and $h (y) = y^{3}$ are continuous at every real number; therefore, by the product of continuous functions theorem, $f (x, y) = 2 x^{2} y^{3}$ is continuous at every point $(x, y)$ in the $x y -plane.$ Furthermore, any constant function is continuous everywhere, so $g (x, y) = 3$ is continuous at every point $(x, y)$ in the $x y -plane.$ Therefore, $f (x, y) = 2 x^{2} y^{3} + 3$ is continuous at every point $(x, y)$ in the $x y -plane.$ Last, $h (x) = x^{4}$ is continuous at every real number $x,$ so by the continuity of composite functions theorem $g (x, y) = {(2 x^{2} y^{3} + 3)}^{4}$ is continuous at every point $(x, y)$ in the $x y -plane.$

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Functions of three or more variables

The limit of a function of three or more variables occurs readily in applications. For example, suppose we have a function $f (x, y, z)$ that gives the temperature at a physical location $(x, y, z)$ in three dimensions. Or perhaps a function $g (x, y, z, t)$ can indicate air pressure at a location $(x, y, z)$ at time $t .$ How can we take a limit at a point in $ℝ^{3} ?$ What does it mean to be continuous at a point in four dimensions?

The answers to these questions rely on extending the concept of a $δ$ disk into more than two dimensions. Then, the ideas of the limit of a function of three or more variables and the continuity of a function of three or more variables are very similar to the definitions given earlier for a function of two variables.

Definition

Let $(x_{0}, y_{0}, z_{0})$ be a point in $ℝ^{3} .$ Then, a $δ$ ball in three dimensions consists of all points in $ℝ^{3}$ lying at a distance of less than $δ$ from $(x_{0}, y_{0}, z_{0})$ —that is,

{(x, y, z) \in ℝ^{3} | \sqrt{{(x - x_{0})}^{2} + {(y - y_{0})}^{2} + {(z - z_{0})}^{2}} < δ} .

To define a $δ$ ball in higher dimensions, add additional terms under the radical to correspond to each additional dimension. For example, given a point $P = (w_{0}, x_{0}, y_{0}, z_{0})$ in $ℝ^{4},$ a $δ$ ball around $P$ can be described by

{(w, x, y, z) \in ℝ^{4} | \sqrt{{(w - w_{0})}^{2} + {(x - x_{0})}^{2} + {(y - y_{0})}^{2} + {(z - z_{0})}^{2}} < δ} .

To show that a limit of a function of three variables exists at a point $(x_{0}, y_{0}, z_{0}),$ it suffices to show that for any point in a $δ$ ball centered at $(x_{0}, y_{0}, z_{0}),$ the value of the function at that point is arbitrarily close to a fixed value (the limit value). All the limit laws for functions of two variables hold for functions of more than two variables as well.

Finding the limit of a function of three variables

Find $lim_{(x, y, z) \to (4, 1, −3)} \frac{x^{2} y - 3 z}{2 x + 5 y - z} .$

Before we can apply the quotient law, we need to verify that the limit of the denominator is nonzero. Using the difference law, the identity law, and the constant law,

\begin{matrix} lim_{(x, y, z) \to (4, 1, −3)} (2 x + 5 y - z) & = 2 (lim_{(x, y, z) \to (4, 1, −3)} x) + 5 (lim_{(x, y, z) \to (4, 1, −3)} y) - (lim_{(x, y, z) \to (4, 1, −3)} z) \\ = 2 (4) + 5 (1) - (−3) \\ = 16. \end{matrix}

Since this is nonzero, we next find the limit of the numerator. Using the product law, difference law, constant multiple law, and identity law,

\begin{matrix} lim_{(x, y, z) \to (4, 1, −3)} (x^{2} y - 3 z) & = {(lim_{(x, y, z) \to (4, 1, −3)} x)}^{2} (lim_{(x, y, z) \to (4, 1, −3)} y) - 3 lim_{(x, y, z) \to (4, 1, −3)} z \\ = (4^{2}) (1) - 3 (−3) \\ = 16 + 9 \\ = 25 . \end{matrix}

Last, applying the quotient law:

\begin{matrix} lim_{(x, y, z) \to (4, 1, −3)} \frac{x^{2} y - 3 z}{2 x + 5 y - z} & = \frac{lim_{(x, y, z) \to (4, 1, −3)} (x^{2} y - 3 z)}{lim_{(x, y, z) \to (4, 1, −3)} (2 x + 5 y - z)} \\ = \frac{25}{16} . \end{matrix}

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Find $lim_{(x, y, z) \to (4, −1, 3)} \sqrt{13 - x^{2} - 2 y^{2} + z^{2}} .$

$lim_{(x, y, z) \to (4, −1, 3)} \sqrt{13 - x^{2} - 2 y^{2} + z^{2}} = 2$

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Key concepts

To study limits and continuity for functions of two variables, we use a $δ$ disk centered around a given point.
A function of several variables has a limit if for any point in a $δ$ ball centered at a point $P,$ the value of the function at that point is arbitrarily close to a fixed value (the limit value).
The limit laws established for a function of one variable have natural extensions to functions of more than one variable.
A function of two variables is continuous at a point if the limit exists at that point, the function exists at that point, and the limit and function are equal at that point.

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Source: OpenStax, Calculus volume 3. OpenStax CNX. Feb 05, 2016 Download for free at http://legacy.cnx.org/content/col11966/1.2

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