Suppose 𝐮 and 𝐯 are vector functions that possess limits as t →a and let c be a constant. Prove

step 1 We are asked to prove properties of limits of vector functions. In part A, we're given two vecto

Suppose 𝐮 and 𝐯 are vector functions that possess limits as...

Suppose $\mathbf{u}$ and $\mathbf{v}$ are vector functions that possess limits as $t \rightarrow a$ and let $c$ be a constant. Prove the following properties of limits. (a) $$\lim _{t \rightarrow a}[\mathbf{u}(t)+\mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t)+\lim _{t \rightarrow a} \mathbf{v}(t)$$ (b) $$\lim _{r \rightarrow a} c \mathbf{u}(t)=c \lim _{t \rightarrow a} \mathbf{u}(t)$$ (c) $$\lim _{t \rightarrow a}[\mathbf{u}(t) \cdot \mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t) \cdot \lim _{t \rightarrow a} \mathbf{v}(t)$$ (d) $$\lim _{t \rightarrow a}[\mathbf{u}(t) \times \mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t) \times \lim _{t \rightarrow a} \mathbf{v}(t)$$

Suppose $\mathbf{u}$ and $\mathbf{v}$ are vector functions that possess limits as
$t \rightarrow a$ and let $c$ be a constant. Prove the following properties of limits.

(a) $$\lim _{t \rightarrow a}[\mathbf{u}(t)+\mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t)+\lim _{t \rightarrow a} \mathbf{v}(t)$$
(b) $$\lim _{r \rightarrow a} c \mathbf{u}(t)=c \lim _{t \rightarrow a} \mathbf{u}(t)$$
(c) $$\lim _{t \rightarrow a}[\mathbf{u}(t) \cdot \mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t) \cdot \lim _{t \rightarrow a} \mathbf{v}(t)$$
(d) $$\lim _{t \rightarrow a}[\mathbf{u}(t) \times \mathbf{v}(t)]=\lim _{t \rightarrow a} \mathbf{u}(t) \times \lim _{t \rightarrow a} \mathbf{v}(t)$$

Key Concepts

Limit of a Vector Function

A vector function’s limit is defined component?wise. That is, a vector function u(t) approaches a limit L as t approaches a if and only if each of its component functions converges to the corresponding component of L. This concept allows us to use techniques from real analysis on each component separately.

Algebraic Limit Laws

Algebraic limit laws state that if limits exist, then the limit of a sum is the sum of the limits, and the limit of a scalar multiple is the scalar multiplied by the limit. These rules are the foundation for manipulating limits and are directly applicable to vector functions due to their component-wise definition.

Scalar Multiplication Limit

Scalar multiplication is continuous, meaning that multiplying a vector function by a constant and then taking the limit is equivalent to taking the limit of the vector function first and then multiplying by the constant. This follows directly from the algebraic limit laws and the structure of real numbers.

Dot Product Limit

The dot product of two vector functions is computed as the sum of the products of their corresponding components. Since limits distribute over finite sums and products (when limits exist), the limit of the dot product is the dot product of the limits of the individual functions.

Cross Product Limit

Similar to the dot product, the cross product of two vector functions involves operations on their components. Because the limit of a combination of sums and products of functions is the combination of their limits (assuming all limits exist), the cross product of the limits equals the limit of the cross products.

Transcript

00:02 We are asked to prove properties of limits of vector functions.

00:12 In part a, we're given two vector functions u and b that possess limits as t approaches a.

00:22 We want to show that the limit as t approaches a of ut plus vt is the same as the limit as t approaches a of ut plus the limit as t approaches a of v2.

00:36 Prove this.

00:37 I'll start on the left -hand side.

00:39 We have this limit as t approaches a of ut plus vt.

00:52 This is going to be the same as the limit as t approaches a of the vector.

01:00 And you have that ut was u1, u2, u3t, and vt is u1t, or vt, v2t, v2t, v2t, v3t.

01:09 This is the limit as t approaches a on the vector u1t plus v1 t u2 t plus v2 t and u3 t plus v3 t and we have that since the limit is t approaches a of u of t and the limit is t approaches a of v of t exists we have the limit is t approaches a on u1 of t the limit is t approaches a of u 2 of t and the limit is t approaches a on the 3 of t all exist, likewise for each of the components of v of t.

02:09 And so we have that each of the sums, u1 of t plus v1 of t and so on, has a limit as t approaches a.

02:19 And therefore, we have that the limit as t approaches a of this vector exists and is given as the vector whose components are, limits as t approaches a, of the original functions.

02:42 This is limit as 2 .5 of u1t plus v1t.

02:46 The limit is t approaches a on u2t plus v2t.

02:57 And the limit as t approaches a of u2 or u3t plus v3t.

03:14 And by the addivity of real value functions, limits of real value functions, this is going to be equal to the sums of, see, the limit as t approaches a on u1 of t, the limit as t approaches a, u2t, and the limit as t approaches a of u3t.

03:49 And i'm writing as two different vectors, this is this plus the vector, the limit as t approaches a, of v1t, limit as t approaches a of v2t, and a limit as t approaches a of v3t.

04:13 We recognize that since the limits of u and v, v, t, approaches, a, this is the same as the limit as t approaches a of ut, plus the limit as t approaches a of ut, plus the limit as t approaches a of ut, a of vt, which is what we wanted to show for part a.

04:38 In part b, we're asked to show that the limit as t approaches a of c u .t is equal to the limit as you approach as t approaches a of u .t times c.

04:53 So i'll start on the left -hand side, part b.

04:59 The limit as t approaches a of c u .t.

05:05 This is the same as the limit as t approaches a of the vector c .u1t, c .u2t, cu3t.

05:23 And then since the limit as t approaches a of ut exists, it follows the limit as t a purchase a of each of the components of u exists.

05:35 And so we have that by the properties of limits of real value functions, the limits as t a purchase of c times each component exists, and therefore it follows the limit as t.

05:59 Purchase on this vector exists, since each of the limits of its components exists, and is given by the vector whose components are the limits of the component functions of the original vector.

06:19 So this is the limit as t approaches a of c u1t, limit as t approaches a of c u2t, and the limit as t approaches a of cu3t...

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