Part 3/6:
Visualization of Limits
To visualize this relationship, imagine a three-dimensional coordinate space where we visualize the vector ( \mathbf{l} ) as a line in this space. As ( t ) approaches ( a ), the vector function ( \mathbf{r}(t) ) draws nearer to this line (or vector). This geometric interpretation helps clarify how limit processes in vector calculus can describe trajectories and convergence of vectors.
Properties of Limits
Similar to limits of real-valued functions, limits of vector functions obey analogous rules. For instance, the limit of a sum or scalar multiple of vector functions follows predictable patterns, just like in the one-dimensional case.