{"id":1066,"date":"2025-10-15T13:55:26","date_gmt":"2025-10-15T13:55:26","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&p=1066"},"modified":"2025-10-23T13:58:33","modified_gmt":"2025-10-23T13:58:33","slug":"limits-and-continuity-learn-it-4","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/limits-and-continuity-learn-it-4\/","title":{"raw":"Limits and Continuity: Learn It 4","rendered":"Limits and Continuity: Learn It 4"},"content":{"raw":"

Functions of Three or More Variables<\/h2>\r\n

The limit of a function of three or more variables occurs readily in applications. For example, suppose we have a function [latex]f\\,(x,\\ y,\\ z)[\/latex]\u00a0that gives the temperature at a physical location [latex](x,\\ y,\\ z)[\/latex]\u00a0in three dimensions. Or perhaps a function [latex]g\\,(x,\\ y,\\ z,\\ t)[\/latex]\u00a0can indicate air pressure at a location\u00a0[latex](x,\\ y,\\ z)[\/latex] at time [latex]t[\/latex].<\/p>\r\n

How can we take a limit at a point in [latex]\\mathbb{R}^{3}[\/latex]? What does it mean to be continuous at a point in four dimensions?<\/p>\r\n

The answers to these questions rely on extending the concept of a [latex]\\delta[\/latex] disk into more than two dimensions. Then, the ideas of limits and continuity for functions of three or more variables follow naturally from the definitions for two variables.<\/p>\r\n\r\n

\r\n

[latex]\\delta[\/latex]\u00a0ball in higher dimensions<\/h3>\r\nLet [latex](x_0,\\ y_0,\\ z_0)[\/latex] be a point in [latex]\\mathbb{R}^{3}[\/latex]. A [latex]\\delta[\/latex] ball<\/strong> in three dimensions consists of all points in [latex]\\mathbb{R}^{3}[\/latex]\u00a0lying at a distance of less than [latex]\\delta[\/latex] from\u00a0[latex](x_0,\\ y_0,\\ z_0)[\/latex]:\r\n
[latex]\\large{\\big\\{(x,\\ y,\\ z)\\in\\mathbb{R}^{3}\\big|\\sqrt{(x-x_0)^{2}+(y-y_0)^{2}+(z-z_0)^{2}}\\,<\\delta\\big\\}}[\/latex].<\/div>\r\n \r\n\r\nFor higher dimensions, add additional terms under the radical to correspond to each additional dimension.\r\n\r\n<\/section>
For example, given a point [latex]P=(w_0,\\ x_0,\\ y_0,\\ z_0)[\/latex] in [latex]\\mathbb{R}^{4}[\/latex], a [latex]\\delta[\/latex] ball around [latex]P[\/latex] can be described by\r\n
[latex]\\large{\\big\\{(w,\\ x,\\ y,\\ z)\\in\\mathbb{R}^{4}\\big|\\sqrt{(w-w_0)^{2}+(x-x_0)^{2}+(y-y_0)^{2}+(z-z_0)^{2}}\\,<\\delta\\big\\}}[\/latex].<\/div>\r\n<\/section>To show that a limit of a function of three variables exists at a point [latex](x_0,\\ y_0,\\ z_0)[\/latex], we show that for any point in a [latex]\\delta[\/latex] ball centered at [latex](x_0,\\ y_0,\\ z_0)[\/latex], 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.\r\n\r\n
Find [latex]\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}\\frac{x^{2}y-3z}{2x+5y-z}.[\/latex][reveal-answer q=\"fs-id1647993733114\"]Show Solution[\/reveal-answer]\r\n[hidden-answer a=\"fs-id1647993733114\"]\r\n\r\nBefore 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,\r\n
[latex]\\begin{array}{ccc}\\hfill {\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}(2x+5y-z)} & =\\hfill & {2\\left(\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}x\\right)+5\\left(\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}y\\right)-\\left(\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}z\\right)} \\hfill \\\\ \\hfill &=\\hfill & {2(4)+5(1)-(-3)} \\hfill \\\\ \\hfill & =\\hfill & {16.}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n \r\n\r\nSince this\u00a0is nonzero, we next find the limit of the numerator. Using the product law, difference law, constant multiple law, and identity law,\r\n
[latex]\\begin{array}{ccc}\\hfill {\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}(x^{2}y-3z)} & =\\hfill & {\\left(\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}x\\right)^{2}\\,\\left(\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}y\\right)-3\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}z}\\hfill \\\\ \\hfill & =\\hfill & {(4^{2})(1)-3(-3)}\\hfill \\\\ \\hfill & =\\hfill & {16+9}\\hfill \\\\ \\hfill & =\\hfill & {25.}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n \r\n\r\nLast, applying the quotient law:\r\n
[latex]\\begin{array}{ccc}\\hfill {\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}\\frac{x^{2}y-3z}{2x+5y-z}} &=\\hfill & {\\frac{\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}(x^{2}y-3z)}{\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}(2x+5y-z)}}\\hfill \\\\ \\hfill & =\\hfill & {\\frac{25}{16}.} \\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n[\/hidden-answer]\r\n\r\n<\/section>","rendered":"

Functions of Three or More Variables<\/h2>\n

The limit of a function of three or more variables occurs readily in applications. For example, suppose we have a function [latex]f\\,(x,\\ y,\\ z)[\/latex]\u00a0that gives the temperature at a physical location [latex](x,\\ y,\\ z)[\/latex]\u00a0in three dimensions. Or perhaps a function [latex]g\\,(x,\\ y,\\ z,\\ t)[\/latex]\u00a0can indicate air pressure at a location\u00a0[latex](x,\\ y,\\ z)[\/latex] at time [latex]t[\/latex].<\/p>\n

How can we take a limit at a point in [latex]\\mathbb{R}^{3}[\/latex]? What does it mean to be continuous at a point in four dimensions?<\/p>\n

The answers to these questions rely on extending the concept of a [latex]\\delta[\/latex] disk into more than two dimensions. Then, the ideas of limits and continuity for functions of three or more variables follow naturally from the definitions for two variables.<\/p>\n

\n

[latex]\\delta[\/latex]\u00a0ball in higher dimensions<\/h3>\n

Let [latex](x_0,\\ y_0,\\ z_0)[\/latex] be a point in [latex]\\mathbb{R}^{3}[\/latex]. A [latex]\\delta[\/latex] ball<\/strong> in three dimensions consists of all points in [latex]\\mathbb{R}^{3}[\/latex]\u00a0lying at a distance of less than [latex]\\delta[\/latex] from\u00a0[latex](x_0,\\ y_0,\\ z_0)[\/latex]:<\/p>\n

[latex]\\large{\\big\\{(x,\\ y,\\ z)\\in\\mathbb{R}^{3}\\big|\\sqrt{(x-x_0)^{2}+(y-y_0)^{2}+(z-z_0)^{2}}\\,<\\delta\\big\\}}[\/latex].<\/div>\n

 <\/p>\n

For higher dimensions, add additional terms under the radical to correspond to each additional dimension.<\/p>\n<\/section>\n

For example, given a point [latex]P=(w_0,\\ x_0,\\ y_0,\\ z_0)[\/latex] in [latex]\\mathbb{R}^{4}[\/latex], a [latex]\\delta[\/latex] ball around [latex]P[\/latex] can be described by<\/p>\n
[latex]\\large{\\big\\{(w,\\ x,\\ y,\\ z)\\in\\mathbb{R}^{4}\\big|\\sqrt{(w-w_0)^{2}+(x-x_0)^{2}+(y-y_0)^{2}+(z-z_0)^{2}}\\,<\\delta\\big\\}}[\/latex].<\/div>\n<\/section>\n

To show that a limit of a function of three variables exists at a point [latex](x_0,\\ y_0,\\ z_0)[\/latex], we show that for any point in a [latex]\\delta[\/latex] ball centered at [latex](x_0,\\ y_0,\\ z_0)[\/latex], 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.<\/p>\n

Find [latex]\\displaystyle\\lim_{(x,\\ y,\\ z)\\to(4,\\ 1,\\ -3)}\\frac{x^{2}y-3z}{2x+5y-z}.[\/latex]<\/p>\n