Let 𝐫(t) and 𝐮(t) be vector-valued functions whose limits exist as t →c . Prove that limt →c[𝐫(t

step 1 We've been asked to let R of T and U of T be vector valued functions whose limits exist as T app

Let 𝐫(t) and 𝐮(t) be vector-valued functions whose limits...

Let $\mathbf{r}(t)$ and $\mathbf{u}(t)$ be vector-valued functions whose limits exist as $t \rightarrow c .$ Prove that
$$\lim _{t \rightarrow c}[\mathbf{r}(t) \times \mathbf{u}(t)]=\lim _{t \rightarrow c} \mathbf{r}(t) \times \lim _{t \rightarrow c} \mathbf{u}(t)$$

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Key Concepts

Limits of Vector-Valued Functions

This concept involves understanding that a vector-valued function is essentially a collection of scalar functions—one for each coordinate. The limit of a vector-valued function exists if and only if the limits of all its component functions exist. This framework allows us to handle the convergence of vectors by analyzing each coordinate individually.

Continuity of the Cross Product Operation

The cross product is a bilinear operation defined on vectors in three-dimensional space, and it is continuous with respect to its inputs. This means that if two sequences (or functions) of vectors converge to some limit vectors, then the cross product of the sequences converges to the cross product of the limits. This property is crucial for proving that limit operations can be interchanged with the cross product.

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