Chapter 11: Q. 15 (page 860)

Let $r (t) = (x (t), y (t)), t \in [a, \infty),$ be a vector-valued function, where a is a real number. Under what conditions would the graph of r have a horizontal asymptote as $t \to \infty ?$ Provide an example illustrating your answer.

Short Answer

Expert verified

The vector-valued function has a horizontal asymptote if $\lim_{t \to \pm \infty} y (t) = b$ condition is satisfied.

Step by step solution

Step 1. Given Information.

The given vector-valued function is $r (t) = (x (t), y (t)), t \in [a, \infty) .$

Step 2. Explanation.

The vector-valued function has a horizontal asymptote if $\lim_{t \to \pm \infty} y (t) = b$ condition is satisfied.

So, let's take an example to illustrate the answer,

$Let r (t) = (\frac{1}{t}, \frac{3 t - 2}{t + 1}), t \in [a, \infty) So, y (t) = \frac{3 t - 2}{t + 1} \lim_{t \to \pm \infty} y (t) = \lim_{t \to \pm \infty} \frac{3 t - 2}{t + 1} \lim_{t \to \pm \infty} y (t) = \lim_{t \to \pm \infty} \frac{t (3 - \frac{2}{t})}{t (1 + \frac{1}{t})} \lim_{t \to \pm \infty} y (t) = \frac{3}{1} \lim_{t \to \pm \infty} y (t) = 3$

Hence proved.

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Let $r (t) = (x (t), y (t)), t \in [a, \infty),$ be a vector-valued function, where a is a real number. Under what conditions would the graph of r have a horizontal asymptote as $t \to \infty ?$ Provide an example illustrating your answer.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

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Let r(t)=x(t),y(t),t∈[a,∞),be a vector-valued function, where a is a real number. Under what conditions would the graph of r have a horizontal asymptote as t→∞?Provide an example illustrating your answer.

Short Answer

Step by step solution

Step 1. Given Information.

Step 2. Explanation.

One App. One Place for Learning.

Most popular questions from this chapter

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Study anywhere. Anytime. Across all devices.

Let $r (t) = (x (t), y (t)), t \in [a, \infty),$ be a vector-valued function, where a is a real number. Under what conditions would the graph of r have a horizontal asymptote as $t \to \infty ?$ Provide an example illustrating your answer.