Question

In Exercises \(7-20,\) find the vertical asymptotes (if any) of the function. $$ g(\theta)=\frac{\tan \theta}{\theta} $$

Short answer

The vertical asymptotes of the function \(g(\theta) = \frac{\tan \theta}{\theta}\) are at \(\theta = k\pi + \frac{\pi}{2}\), where \(k\) is an integer.

Step-by-step solution

Identify where the function is undefined

First off, the function is undefined when the denominator is zero, i.e. when \(\theta = 0\). However, to see if this is a vertical asymptote, check the limit as \(\theta\) approaches 0. If the limit is infinity or negative infinity, then there is a vertical asymptote.

Apply L'Hopital's Rule

In this case we have a \(\frac{0}{0}\) condition when \(\theta\) approaches 0. So use L'Hopital's rule, which states that if the limit as x approaches a number c is in the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), the limit can then be found by calculating the limit of the derivative of the numerator divided by the derivative of the denominator. The derivative of \(\tan(\theta)\) is \(\sec^2(\theta)\) and the derivative of \(\theta\) is 1.

Determine the limit at zero

Use L'Hopital's rule to determine the limit at zero by substituting zero into the derivative. Here, we have limit as \(\theta\) approaches 0 of \(\frac{\sec^2(\theta)}{1}\) which equals to 1 since \(\sec(0) = 1\). So the function does not have a vertical asymptote at \(\theta = 0\) because it does not approach infinity or negative infinity.

Identify additional asymptotes

Now, look for additional vertical asymptotes. Recall that the tangent function is undefined at \(\theta = k\pi + \frac{\pi}{2}\), where \(k\) is an integer. Now, divide by \(\theta\). The function is undefined at these points.

Use L'Hopital's Rule for the additional asymptotes

For these additional points, where \(\theta = k\pi + \frac{\pi}{2}\), apply L'Hopital's Rule again by evaluating the limit at these points by finding the derivative of the numerator and the denominator. The limit as \(\theta\) approaches \(k\pi + \frac{\pi}{2}\) of \(\frac{\sec^2(\theta)}{1}\) is undefined, since secant is undefined at these points, which shows that there are vertical asymptotes at these points. Hence, the vertical asymptotes are at \(\theta = k\pi + \frac{\pi}{2}\), where \(k\) is an integer.

Key concepts

L'Hopital's Rule

When you encounter calculus limit problems where the limit of a function as it approaches a specific value is not clear, L'Hopital's Rule often comes to the rescue. It's a powerful piece of mathematical theory which claims that under certain conditions, when a limit results in an indeterminate form like \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \), you can compute the limit by taking derivatives.

To apply L'Hopital's Rule effectively, you must:

• Check that the original limit results in an indeterminate form.
• Find the derivative of the numerator and denominator separately.
• Calculate the limit of the new fraction formed by these derivatives.

If the result is still an indeterminate form, you can repeat the process. Remember to ensure the functions involved are differentiable around the point of interest - this is crucial for applying the rule successfully.

The Tangent Function

In trigonometry, the tangent function is quite unique and is often involved in calculus limit problems. Fundamentally, \( \tan(\theta) \) is the ratio of the opposite side to the adjacent side of a right-angled triangle. In the unit circle context, it represents the y-coordinate divided by the x-coordinate at a certain angle from the positive x-axis.

It's essential to remember that the tangent function:

• Is periodic, with a period of \( \pi \).
• Has vertical asymptotes where the function is undefined, at \( \theta = k\pi + \frac{\pi}{2} \) for integers \(k\).
• Can lead to limit problems that are indeterminate at certain angles, and thus require further analysis, often with L'Hopital's Rule.

Keeping these characteristics in mind helps you understand its behavior and how it can affect the functions it’s a part of.

Calculus Limit Problems

Calculus is replete with limit problems, and grasping these is pivotal in understanding the behavior of functions as they approach specific points. Limits help predict function behavior without direct substitution, which is especially useful when the substitution yields undefined expressions.

Key steps in solving limit problems include:

• Direct substitution, to see if the limit can be easily computed.
• Factoring, to simplify the function and remove indeterminate forms.
• Special rules like L'Hopital's Rule for indeterminate forms.
• Graphical or numerical approaches when analytical methods are complex.

Understanding limits is not just about solving an equation; it's about predicting the trend of functions—how they behave as they inch closer to a specific value.

Undefined Function Behavior

When dealing with functions, you'll occasionally encounter points where they are undefined - places where they have discontinuities or breaks. At those points, functions can exhibit interesting behaviors. Some common forms of undefined behavior are vertical asymptotes and holes.

To investigate these behaviors, you typically:

• Analyze limits to understand the function's approach to the undefined point.
• Look at the broader context or graph of the function for a visual understanding.
• Use algebraic manipulation to see if the undefined nature can be 'resolved', which sometimes reveals removable discontinuities as opposed to permanent ones.

Such analysis of undefined function behavior is crucial for understanding the complexities of a function's graph and predicting its behavior outside the domain of standard definitions.