Determine the following limits.


$\underset{\theta \to {\frac{\pi }{2}}^{+}}{\lim }\frac{1}{3}\tan \theta$

Verified step by step guidance

1. Step 1: Understand the problem. We need to find the limit of the function \( \frac{1}{3} \tan(\theta) \) as \( \theta \) approaches \( \frac{\pi}{2}^+ \). This means we are considering values of \( \theta \) that are slightly greater than \( \frac{\pi}{2} \).
2. Step 2: Analyze the behavior of \( \tan(\theta) \) near \( \frac{\pi}{2} \). The tangent function, \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \), becomes undefined at \( \theta = \frac{\pi}{2} \) because \( \cos(\theta) = 0 \) at this point.
3. Step 3: Consider the right-hand limit. As \( \theta \) approaches \( \frac{\pi}{2}^+ \), \( \cos(\theta) \) approaches 0 from the negative side, making \( \tan(\theta) \) approach \(-\infty \).
4. Step 4: Apply the limit to the function. Since \( \tan(\theta) \to -\infty \) as \( \theta \to \frac{\pi}{2}^+ \), the expression \( \frac{1}{3} \tan(\theta) \) will also approach \(-\infty \).
5. Step 5: Conclude the limit. Therefore, the limit \( \lim_{\theta \to \frac{\pi}{2}^+} \frac{1}{3} \tan(\theta) = -\infty \).

Key Concepts

Limits

Limits are fundamental concepts in calculus that describe the behavior of a function as its input approaches a certain value. They help in understanding how functions behave near points of interest, including points where they may not be defined. In this case, we are examining the limit of the tangent function as the angle approaches π/2 from the right, which is crucial for determining the function's behavior near that point.

Tangent Function

The tangent function, denoted as tan(θ), is a trigonometric function defined as the ratio of the opposite side to the adjacent side in a right triangle. It can also be expressed as sin(θ)/cos(θ). As θ approaches π/2, the cosine of θ approaches zero, causing the tangent function to approach infinity, which is essential for evaluating the limit in the given question.

One-Sided Limits

One-sided limits refer to the value that a function approaches as the input approaches a specific point from one side only, either from the left or the right. In this problem, we are interested in the limit as θ approaches π/2 from the right (denoted as θ → π/2⁺), which is important because the behavior of the tangent function differs when approaching from different directions, particularly near vertical asymptotes.