Please help How do you graph the polar equation r=8sec#theta# ?
See the explanation and graphs.
graph is outside the annular ( circular ) region # 7 < r < 9 ).
Converting to the Cartesian form, using
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To graph the polar equation ( r = 8  \sec(\theta) ), follow these steps:

Plot the asymptotes: The equation ( r = 8  \sec(\theta) ) has vertical asymptotes wherever ( \sec(\theta) ) is undefined. Since ( \sec(\theta) ) is undefined when ( \cos(\theta) = 0 ), there will be vertical asymptotes where ( \cos(\theta) = 0 ), which are ( \theta = \frac{\pi}{2} ) and ( \theta = \frac{3\pi}{2} ).

Determine the behavior near the asymptotes: As ( \theta ) approaches ( \frac{\pi}{2} ) or ( \frac{3\pi}{2} ), ( \sec(\theta) ) approaches ( +\infty ), so ( r ) approaches ( \infty ). Therefore, the graph will approach the vertical asymptotes as ( r ) decreases without bound.

Plot key points: Choose some values of ( \theta ) and compute the corresponding values of ( r ) to plot key points on the graph.

Connect the points: Connect the plotted points smoothly to sketch the graph.
Remember, the graph of polar equations may not always be perfect due to the limitations of plotting tools. However, plotting several points and understanding the behavior near asymptotes will help you visualize the graph accurately.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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