{"id":949,"date":"2025-10-14T16:32:49","date_gmt":"2025-10-14T16:32:49","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/calculus3\/?post_type=chapter&p=949"},"modified":"2025-10-24T13:16:12","modified_gmt":"2025-10-24T13:16:12","slug":"introduction-to-vector-valued-functions-learn-it-3","status":"publish","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/calculus3\/chapter\/introduction-to-vector-valued-functions-learn-it-3\/","title":{"raw":"Introduction to Vector-Valued Functions: Learn It 3","rendered":"Introduction to Vector-Valued Functions: Learn It 3"},"content":{"raw":"
Limits and Continuity of a Vector-Valued Function<\/h2>\r\n
Understanding limits of vector-valued functions is essential for studying their calculus. The process builds directly on what you know about limits of single-variable functions.<\/p>\r\n\r\n\r\n
limit of a vector valued function<\/h3>\r\nA vector-valued function [latex]{\\bf{r}}[\/latex] approaches the limit [latex]\\bf{L}[\/latex] as [latex]t[\/latex] approaches [latex]a[\/latex], written:\r\n
This is the rigorous definition, but in practice we use a simpler approach: calculate the limit of each component function separately.<\/p>\r\n\r\n\r\n
limit of a vector-valued function theorem<\/h3>\r\n
Let [latex]f[\/latex], [latex]g[\/latex], and [latex]h[\/latex] be functions of [latex]t[\/latex].<\/p>\r\n
provided the limits [latex]\\underset{t\\to a}{\\text{lim}}f(t)[\/latex], [latex]\\underset{t\\to a}{\\text{lim}}g(t)[\/latex], and [latex]\\underset{t\\to a}{\\text{lim}}h(t)[\/latex] exist.<\/p>\r\n\r\n<\/section>Think of this as taking the limit \"component by component.\" Each component function follows the same limit rules you learned in Calculus 1.<\/section>In the following example, we show how to calculate the limit of a vector-valued function.\r\n\r\nFor each of the following vector-valued functions, calculate [latex]\\displaystyle\\lim_{t{\\to}3}{\\bf{r}}\\,(t)[\/latex] for\r\n\r\n \t
Use the Limit of a Vector-Valued Function theorem and substitute the value [latex]t=3[\/latex] into the three component expression:\r\n
[latex]\\begin{array}{ccc}\\hfill \\displaystyle\\lim_{t{\\to}3}{\\bf{r}}\\,(t)&=\\hfill&{\\displaystyle\\lim_{t{\\to}3}\\bigg(\\frac{2t-4}{t+1}\\,{\\bf{i}}+\\frac{t}{t^{2}+1}\\,{\\bf{j}}+(4t-3){\\bf{k}}\\bigg)}\\hfill \\\\ \\hfill & =\\hfill & {\\bigg[\\displaystyle\\lim_{t{\\to}3}{(\\frac{2t-4}{t+1})}\\bigg]\\,{\\bf{i}}+\\bigg[\\displaystyle\\lim_{t{\\to}3}{(\\frac{t}{t^{2}+1})}\\bigg]\\,{\\bf{j}}+\\bigg[\\displaystyle\\lim_{t{\\to}3}{(4t-3)}\\bigg]\\,{\\bf{k}}} \\hfill \\\\ \\hfill & =\\hfill & {\\frac{1}{2}\\,{\\bf{i}}+\\frac{3}{10}\\,{\\bf{j}}+9\\,{\\bf{k}}}\\hfill \\\\ \\hfill \\end{array}[\/latex]<\/div>\r\n <\/li>\r\n<\/ol>\r\n[\/hidden-answer]\r\n\r\n<\/div>\r\n<\/section>Now that we can calculate limits of vector-valued functions, we can define what it means for such a function to be continuous.\r\n\r\n\r\n
continuity of a vector-valued function<\/h3>\r\n
Let [latex]f[\/latex], [latex]g[\/latex], and [latex]h[\/latex] be functions of [latex]t[\/latex].<\/p>\r\n
The vector-valued function [latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}[\/latex] is continuous at point [latex]t=a[\/latex] if three conditions hold:<\/p>\r\n\r\n
[latex]\\displaystyle\\lim_{t{\\to}a}{\\bf{r}}\\,(t)={\\bf{r}}\\,(a)[\/latex]<\/li>\r\n<\/ol>\r\nSimilarly, [latex]{\\bf{r}}(t)=f(t){\\bf{i}}+g(t){\\bf{j}}+h(t){\\bf{k}}[\/latex] is continuous at point [latex]t=a[\/latex] if the same three conditions hold.\r\n\r\n<\/section>These three conditions should look familiar\u2014they're the same requirements for continuity you learned with single-variable functions. A vector-valued function is continuous at a point when you can draw its graph through that point without lifting your pencil.<\/section>","rendered":"
Limits and Continuity of a Vector-Valued Function<\/h2>\n
Understanding limits of vector-valued functions is essential for studying their calculus. The process builds directly on what you know about limits of single-variable functions.<\/p>\n\n
limit of a vector valued function<\/h3>\n
A vector-valued function [latex]{\\bf{r}}[\/latex] approaches the limit [latex]\\bf{L}[\/latex] as [latex]t[\/latex] approaches [latex]a[\/latex], written:<\/p>\n
provided the limits [latex]\\underset{t\\to a}{\\text{lim}}f(t)[\/latex], [latex]\\underset{t\\to a}{\\text{lim}}g(t)[\/latex], and [latex]\\underset{t\\to a}{\\text{lim}}h(t)[\/latex] exist.<\/p>\n<\/section>\nThink of this as taking the limit “component by component.” Each component function follows the same limit rules you learned in Calculus 1.<\/section>\n
In the following example, we show how to calculate the limit of a vector-valued function.<\/p>\nFor each of the following vector-valued functions, calculate [latex]\\displaystyle\\lim_{t{\\to}3}{\\bf{r}}\\,(t)[\/latex] for<\/p>\n\n