Chapter H.1, Problem 45E

To determine

To show: the given polar curve has the line $x=2$ as a vertical asymptote

Explanation of Solution

Given:

Curve $r=4+2\sec \theta$

Line $x=2$

Proof:

Rewrite the $sec$ as $cos$

  $\begin{array}{l}r=4+2\sec \theta =4+\frac{2}{\cos \theta }\\ Since x=r\cos \theta , then \cos \theta =\frac{x}{r}\\ r=4+\frac{2}{\frac{x}{r}}\\ r=4+\frac{2r}{x}\\ r-4=\frac{2r}{x}\\ \frac{r-4}{2r}=\frac{1}{x}\\ x=\frac{2r}{r-4}\end{array}$

Evaluate the limit

  $\underset{x\to \pm \infty }{\lim }\frac{2r}{r-4}.\frac{1}{\frac{r}{\frac{1}{r}}}=\underset{x\to \pm \infty }{\lim }\frac{2}{1-\frac{4}{r}}=\frac{2}{1-0}=2$

  $\frac{2r}{r-4}$ , and therefore $xapproaches2asr\to \pm \infty$

This is like a strange sounding backwards description of $x=2$ being a vertical asymptote.

This may be a bit confusing to graph so we could look at the Cartesian graphfirst.

  

  $\begin{array}{l}\underset{x\to \pm \infty }{\lim }\frac{2r}{r-4}.\frac{1}{\frac{r}{\frac{1}{r}}}=\underset{x\to \pm \infty }{\lim }\frac{2}{1-\frac{4}{r}}=\frac{2}{1-0}=2\\ sec\theta isundefinedat\theta =\frac{\pi }{2}+n\pi (nisaninteger)\end{array}$

Start at $\left(r,\theta =\left(6,0\right)\right)$ . As $\theta \to \frac{\pi }{2},$

  $r$ increases rapidly and the curveapproaches $x=2$ , as we found in the beginning of this problem. Youmay be wondering why if $\theta$ is supposed to be close to $\frac{\pi }{2}$ then why the curveseems to be over there towards the right. It is difficult to wrap your headaround, but it's just something you have to think about with $r$ approaching $\infty$ while $\theta$ is not quite $\frac{\pi }{2}$ .

When $\theta$ is past $\frac{\pi }{2}$ the $r$ values are negative and the curve comes up from the other side of $x=2$ . Makes a small loop during the positive part ofthe curve between $\frac{\pi }{2}and\frac{3\pi }{2}$ in the Cartesian graph, then when $r$ goes backdown to $-\infty$ the polar curve approaches $x=2$ from the left moving upwards.

Between $\frac{3\pi }{2}$ and $2\pi$ the curve comes up from below on the other side of $x=2$ to complete one full figure.

  

Conclusion:

Thus, the given polar curve has the line $x=2$ as a vertical asymptote is proved.