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

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Probability and Statistics

Theoretical and Mathematical Physics

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Logic and Functions

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