Chapter 11: Q. 57 (page 862)

Let $r_{1} (t) a n d r_{2} (t)$ be continuous vector functions with two components, and let $t_{0}$ be a point in the domains of both $r_{1} a n d r_{2}$ . Prove that
localid="1649617407427" $lim_{t \to t_{0}} (r_{1} (t) \cdot r_{2} (t)) = r_{1} (t_{0}) \cdot r_{2} (t_{0})$ .

Short Answer

Expert verified

Ans: It is proved that $lim_{t \to t_{0}} (r_{1} (t) \cdot r_{2} (t)) = r_{1} (t_{0}) \cdot r_{2} (t_{0})$

Step by step solution

Step 1. Given information.

given,

$lim_{t \to t_{0}} (r_{1} (t) \cdot r_{2} (t)) = r_{1} (t_{0}) \cdot r_{2} (t_{0})$

Step 2. Solution

Consider two continuous vector functions with two components. Let

$r_{1} (t) = <x_{1} (t), y_{1} (t)> and r_{2} (t) = <x_{2} (t), y_{2} (t)>$

The objective is to prove that

$lim_{t \to t_{0}} (r_{1} (t) \cdot r_{2} (t)) = r_{1} (t_{0}) \cdot r_{2} (t_{0})$

Where $t_{0}$ belongs to the domains of $r_{1} a n d r_{2}$ .

Since $r_{1} (t) a n d r_{2} (t)$ are continuous, their components $x_{1} (t), y_{1} (t), x_{2} (t) and y_{2} (t)$ are continuous.

Thus product and sum of the continuous functions are also continuous.

Step 3. Consider

$\begin{aligned} r_{1} (t) \cdot r_{2} (t) = <x_{1} (t), y_{1} (t)> \cdot <x_{2} (t), y_{2} (t)> \\ = x_{1} (t) x_{2} (t) + y_{1} (t) y_{2} (t) \end{aligned}$

is continuous

Therefore, the function $r_{1} (t) . r_{2} (t)$ is continuous

Consider

$\begin{aligned} lim_{t \to t_{0}} (r_{1} (t) \cdot r_{2} (t)) = lim_{t \to t_{0}} (x_{1} (t) x_{2} (t) + y_{1} (t) y_{2} (t)) \\ = lim_{t \to t_{0}} x_{1} (t) lim_{t \to t_{0}} x_{2} (t) + lim_{t \to t_{0}} y_{1} (t) lim_{t \to t_{0}} y_{2} (t) \\ = x_{1} (t_{0}) x_{2} (t_{0}) + y_{1} (t_{0}) y_{2} (t_{0}) definition of continuous function \\ = r_{1} (t_{0}) \cdot r_{2} (t_{0}) \\ Thus lim_{t \to t_{0}} (r_{1} (t) \cdot r_{2} (k)) = r_{1} (t_{0}) \cdot r_{2} (t_{0}) \end{aligned}$

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Let $r_{1} (t) a n d r_{2} (t)$ be continuous vector functions with two components, and let $t_{0}$ be a point in the domains of both $r_{1} a n d r_{2}$ . Prove that
localid="1649617407427" $lim_{t \to t_{0}} (r_{1} (t) \cdot r_{2} (t)) = r_{1} (t_{0}) \cdot r_{2} (t_{0})$ .

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solution

Step 3. Consider

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Let r1(t)andr2(t) be continuous vector functions with two components, and let t0be a point in the domains of both r1andr2. Prove that localid="1649617407427" limt→t0 r1(t)⋅r2(t)=r1t0⋅r2t0.

Short Answer

Step by step solution

Step 1. Given information.

Step 2. Solution

Step 3. Consider

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Applied Mathematics

Geometry

Probability and Statistics

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Decision Maths

Study anywhere. Anytime. Across all devices.

Let $r_{1} (t) a n d r_{2} (t)$ be continuous vector functions with two components, and let $t_{0}$ be a point in the domains of both $r_{1} a n d r_{2}$ . Prove that
localid="1649617407427" $lim_{t \to t_{0}} (r_{1} (t) \cdot r_{2} (t)) = r_{1} (t_{0}) \cdot r_{2} (t_{0})$ .