Part 1/6:
Understanding the Limit of a Vector Function
The concept of limits is fundamental to understanding calculus, not just for real-valued functions but also for vector functions. In this article, we will explore how the limit of a vector function is defined, how it relates to its component functions, and delve into some examples for clearer understanding.
Definition of Limits for Vector Functions
The limit of a vector function ( \mathbf{r}(t) ) can be defined by examining the limits of its individual components. Specifically, if a vector function comprises three component functions ( f(t), g(t), ) and ( h(t) ) in three-dimensional space, then the limit as ( t ) approaches ( a ) can be expressed as follows:
[
\lim_{t \to a} \mathbf{r}(t) = \begin{pmatrix}