Question

Find the limits. $$ \lim _{\theta \rightarrow(\pi / 2)^{+}} \frac{\pi \theta}{\cos \theta} $$

Short answer

The limit is \(-\infty\).

Step-by-step solution

Identify the problem

We need to find the limit of the function \( \frac{\pi \theta}{\cos \theta} \) as \( \theta \) approaches \( \frac{\pi}{2} \) from the right, denoted as \( \theta \rightarrow (\frac{\pi}{2})^{+} \). This means we approach \( \frac{\pi}{2} \) with values larger than it.

Understand the behavior of \( \cos \theta \)

As \( \theta \rightarrow (\frac{\pi}{2})^{+} \), the value of \( \cos \theta \) approaches 0 from the negative side because cosine changes sign at \( \frac{\pi}{2} \).

Assess implications on the function

Given that \( \cos \theta \rightarrow 0^- \) (negative zero), the denominator of our function \( \frac{\pi \theta}{\cos \theta} \) approaches zero negatively, causing the entire fraction to tend towards negative infinity.

Apply knowledge of limits

Given the behavior of \( \cos \theta \) as it approaches zero from the negative side, the function \( \frac{\pi \theta}{\cos \theta} \) grows without bound in the negative direction. Thus, \( \lim _{\theta \rightarrow (\pi / 2)^{+}} \frac{\pi \theta}{\cos \theta} = -\infty \).

Key concepts

Trigonometric Limits

Trigonometric limits are essential when dealing with functions that include trigonometric expressions such as sine, cosine, and tangent. These functions have unique properties and behaviors especially near certain critical angles like \( \frac{\pi}{2} \), \( \pi \), and \( \frac{3\pi}{2} \). Knowing how to evaluate these limits helps in understanding the behavior of trigonometric functions, which is crucial in calculus.
For the exercise, the function is \( \frac{\pi \theta}{\cos \theta} \), where \( \theta \) approaches \( \frac{\pi}{2} \) from the right. Here, cosine moves towards zero, which demands attention as division by zero implies the limit does not exist in its standard form.
The presence of \( \cos \theta \) in the denominator signifies that at \( \theta = \frac{\pi}{2} \), the function may encounter problems, specifically, undefined behavior. However, by understanding the directional approach towards zero, we can predict the behavior of such functions, allowing us to determine whether a particular direction (positive or negative) is predominated as the value approaches the incrementally small numbers around zero.

One-Sided Limits

One-sided limits come into play when examining how a function behaves as it approaches a particular point from one specific side. This is different from a two-sided limit where the approach is made from both sides. For the problem, we are interested in the function \( \frac{\pi \theta}{\cos \theta} \) as \( \theta \) approaches \( \frac{\pi}{2} \) from the positive side (i.e., \( (\pi/2)^{+} \)).
As \( \theta \) approaches \( \frac{\pi}{2}^{+} \), the cosine function \( \cos \theta \) tends towards zero from the negative side. Understanding one-sided limits is key to identifying cases where the behavior dramatically changes as is evident in our exercise.
Dealing with one-sided limits requires us to focus on the path \( \theta \) takes, because approaching from a larger value impacts the trigonometric function’s behavior distinctly. In this setting, knowledge of trigonometrical identities and behavior near certain key points is useful, enabling more accurate evaluation of the limit.

Behavior of Trigonometric Functions Near Asymptotes

Trigonometric functions exhibit distinctive behavior near their asymptotes. For instance, cosine and sine oscillate between -1 and 1, but they have points, often asymptotes, where their behavior sharply changes. These are points like \( \frac{\pi}{2}, \pi,\frac{3\pi}{2} \) where trigonometric functions may become undefined or diverge.
In the exercise, as \( \theta \rightarrow (\frac{\pi}{2})^{+} \), \( \cos \theta \) nears zero, causing the term \( \frac{1}{\cos \theta} \) in the function to diverge towards either positive or negative infinity. In this case, because it's approaching negative zero, our limit evaluates to \(-\infty\).
This tendency towards infinity, known technically as divergence, especially near asymptotes, is an indicative property of trigonometric functions. Recognizing how these function behaviors are impacted by proximity to asymptotes like zero or points where undefined divisions occur is vital understanding for managing calculus problems dealing with such functions.