   Chapter 12.1, Problem 84E

Chapter
Section
Textbook Problem

# ProofLet r ( t ) and u ( t ) be vector-valued functions whose limits exist as t → c . Prove that lim t → c [ r ( t ) ⋅ u ( t ) ] = lim t → c r ( t ) ⋅ lim t → c u ( t ) .

To determine

To prove: The limit limxc[r(t)u(t)]=limxcr(t)limxcu(t).

Explanation

Given:

The vector-valued functions r(t) and u(t) whose limit exists as tc.

Formula used:

The dot product of two vectors:

(ai+bj+ck)(pi+qj+rk)=ap+bq+cr.

Proof:

Let r(t)=a(t)i+b(t)j+c(t)k and u(t)=d(t)i+e(t)j+f(t)k.

Then,

r(t)u(t)=[a(t)i+b(t)j+c(t)k][d(t)i+e(t)j+f(t)k]=a(t)d(t)+b(t)e(t)+c(t)f(t)

Now,

limxcr(t)u(t)=limxc[a(t)d(t)+b(t)e(t)+

### Still sussing out bartleby?

Check out a sample textbook solution.

See a sample solution

#### The Solution to Your Study Problems

Bartleby provides explanations to thousands of textbook problems written by our experts, many with advanced degrees!

Get Started

#### Expand each expression in Exercises 122. (x+1x)2

Finite Mathematics and Applied Calculus (MindTap Course List)

#### In problems 24-26, find the intercepts and graph. 25.

Mathematical Applications for the Management, Life, and Social Sciences

#### Multiply: (+5)(3)

Elementary Technical Mathematics 