Oh wow thanks for the update. PS. I see that it started working?

Part 1/6:

Analyzing Vector Function Limits

In the realm of calculus, proving properties of limits for vector functions can follow similar principles as those for real-valued functions. In this article, we'll explore the properties of limits concerning vector functions: addition, scalar multiplication, dot product, and cross product.

Property of Addition and Subtraction

To begin, we consider two vector functions, ( \mathbf{u}(t) ) and ( \mathbf{v}(t) ), each possessing limits as ( t ) approaches ( a ). The first property to prove is that the limit of the sum (or difference) of two vector functions equals the sum (or difference) of their limits.

Mathematically, this can be expressed as:

[

Part 2/6:

\lim_{t \to a} \left(\mathbf{u}(t) + \mathbf{v}(t)\right) = \lim_{t \to a} \mathbf{u}(t) + \lim_{t \to a} \mathbf{v}(t)

]

To demonstrate this, we first expand the vector functions into their respective components in 3D space:

[

\mathbf{u}(t) = \begin{pmatrix} u_1(t) \ u_2(t) \ u_3(t) \end{pmatrix}, \quad \mathbf{v}(t) = \begin{pmatrix} v_1(t) \ v_2(t) \ v_3(t) \end{pmatrix}

]

By applying definition one of limits which states that, if the limit of the vector function exists, then the limits of the individual components also exist. Therefore, we break down the limit of the sum into components and verify:

[

\lim_{t \to a} \left( u_1(t) + v_1(t) \right) + \lim_{t \to a} \left( u_2(t) + v_2(t) \right) + \lim_{t \to a} \left( u_3(t) + v_3(t) \right)

]

Part 3/6:

This corresponds to the addition of limits of the individual components. Hence, we have validated that the limit of the sum of vector functions corresponds with the sum of their individual limits.

Property of Scalar Multiplication

Next, we explore the property concerning scalar multiplication. Specifically, the limit of a vector function multiplied by a constant scalar ( c ) can be established:

[

\lim_{t \to a} (c \cdot \mathbf{u}(t)) = c \cdot \lim_{t \to a} \mathbf{u}(t)

]

In a similar approach, we can express ( \mathbf{u}(t) ) in terms of its components and apply the limits to each component independently, thus demonstrating that the scalar can be factored out due to its independence from the variable ( t ):

[

Part 4/6:

= c \left( \lim_{t \to a} u_1(t), \lim_{t \to a} u_2(t), \lim_{t \to a} u_3(t) \right)

]

This shows the consistency of scalar multiplication with the limits of vector functions.

Property of Dot Product

The third property investigates the dot product of two vector functions:

[

\lim_{t \to a} \left(\mathbf{u}(t) \cdot \mathbf{v}(t)\right) = \lim_{t \to a} \mathbf{u}(t) \cdot \lim_{t \to a} \mathbf{v}(t)

]

Following the same strategy, we will expand both vector functions to their components and apply the limit to each component:

[

\lim_{t \to a} (u_1(t) \cdot v_1(t)) + \lim_{t \to a} (u_2(t) \cdot v_2(t)) + \lim_{t \to a} (u_3(t) \cdot v_3(t))

]

Part 5/6:

Once again demonstrating that the limit of the dot product of two vector functions equals the dot product of the limits of each vector function.

Property of Cross Product

Lastly, we delve into the limits concerning the cross product of vector functions:

[

\lim_{t \to a} \left(\mathbf{u}(t) \times \mathbf{v}(t)\right) = \lim_{t \to a} \mathbf{u}(t) \times \lim_{t \to a} \mathbf{v}(t)

]

Applying similar principles, we can expand the equation using the determinant method for vector cross products and verify it through the individual limits of vector components as well, thus arriving at:

[

\lim_{t \to a} (u_1(t) \cdot v_2(t) - u_2(t) \cdot v_1(t), u_3(t) \cdot v_1(t) - u_1(t) \cdot v_3(t), u_1(t) \cdot v_2(t) - u_2(t) \cdot v_1(t))

]

Part 6/6:

Each element reflects the transformations required by the mechanics of the cross product.

Conclusion

Through these proofs, we not only reaffirm the traditional properties of limits but also expand them into the context of vector functions. Each property illustrated above retains its validity by carefully analyzing the components of vector functions. This fundamental understanding serves as a cornerstone in higher-dimensional calculus and provides a functional foundation for analyzing complex vector behaviors.