Q. 15 - Calculus

Question

Let $r\left(t\right)=\left(x\left(t\right), y\left(t\right)\right), t\in [a,\infty ),$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.

Step-by-Step Solution

Answer

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

Step 1. Given Information.

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

Step 2. Explanation.

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

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

$$\begin{gathered} \mathrm{Let} r\left(t\right)=\left(\frac{1}{t},\frac{3t-2}{t+1}\right), t\in [a,\infty ) \\ \mathrm{So}, y\left(t\right)=\frac{3t-2}{t+1} \\ \underset{t\to \pm \infty }{\lim }y\left(t\right)=\underset{t\to \pm \infty }{\lim }\frac{3t-2}{t+1} \\ \underset{t\to \pm \infty }{\lim }y\left(t\right)=\underset{t\to \pm \infty }{\lim }\frac{t\left(3-{\displaystyle \frac{2}{t}}\right)}{t\left(1+{\displaystyle \frac{1}{t}}\right)} \\ \underset{t\to \pm \infty }{\lim }y\left(t\right)=\frac{3}{1} \\ \underset{t\to \pm \infty }{\lim }y\left(t\right)=3 \end{gathered}$$

Hence proved.