Problem 12 - Calculus III

Question

Find the limit of the following vector-valued functions at the indicated value of \(t\). $$ \lim _{t \rightarrow \pi / 6}\left\langle\cos ^{2} t, \sin ^{2} t, 1\right\rangle $$

Step-by-Step Solution

Answer

The limit of the vector-valued function is \( \langle \frac{3}{4}, \frac{1}{4}, 1 \rangle \).

Step 1: Identify the Limit Components

The problem requires us to take the limit of the vector-valued function as the variable \( t \) approaches \( \frac{\pi}{6} \). The components of the vector are \( \cos^2 t \), \( \sin^2 t \), and \( 1 \). We will evaluate each component separately.

Step 2: Calculate Limit of First Component

The first component of the vector is \( \cos^2 t \). Substitute \( t = \frac{\pi}{6} \) into \( \cos^2 t \):\[ \cos \left(\frac{\pi}{6}\right) = \frac{\sqrt{3}}{2} \rightarrow \cos^2 \left(\frac{\pi}{6}\right) = \left( \frac{\sqrt{3}}{2} \right)^2 = \frac{3}{4} \].

Step 3: Calculate Limit of Second Component

The second component is \( \sin^2 t \). Substitute \( t = \frac{\pi}{6} \) into \( \sin^2 t \):\[ \sin \left(\frac{\pi}{6}\right) = \frac{1}{2} \rightarrow \sin^2 \left(\frac{\pi}{6}\right) = \left( \frac{1}{2} \right)^2 = \frac{1}{4} \].

Step 4: Evaluate the Constant Component

The third component is the constant \( 1 \). The limit of a constant is the constant itself, so this remains \( 1 \).

Step 5: Combine Limits into a Vector

Now, we combine the limits of each component vector to form the final limit vector:\[ \langle \frac{3}{4}, \frac{1}{4}, 1 \rangle \].

Key Concepts

Limits

In calculus, the concept of limits is fundamental for understanding how functions behave as they close in on a particular point. When dealing with vector-valued functions,these limits involve breaking down the vector into its individual components and finding the limit for each piece separately.
This procedure becomes especially important when vectors include trigonometric functions, constants, and more complicated expressions.

In our exercise, we have a vector \[ \langle \cos^2 t, \sin^2 t, 1 \rangle \] and are asked to find its limit as \[ t \to \frac{\pi}{6} \] To do this, let's:

• Substitute the target value into each component one at a time.
• Calculate the result for each component limit independently.
• Reassemble these results into a new vector to find the overall limit value.

Working through these steps ensures that we're systematically evaluating the vector's behavior and structure as it approaches a specific point.
Understanding limits is not just about numbers or points. It’s also about comprehending the nature and direction of vectors and functions.

Trigonometric functions

Trigonometric functions like sine and cosine are crucial when working with vector-valued functions.They describe angular relationships and periodic phenomena in mathematics. In problems involving limits, inserting a value of the variable often transforms them into specific numerical results.
Consider the components in our exercise:

• \( \cos^2 t \) involves taking the cosine of an angle squared.
• \( \sin^2 t \) is the sine of an angle squared.

By calculating these at \( t = \frac{\pi}{6} \)- The cosine value becomes \( \frac{\sqrt{3}}{2} \),- Squaring it gives a limit component value of \( \frac{3}{4} \).Similarly for sine:- The sine value becomes \( \frac{1}{2} \)- Squaring it gives a limit component value of \( \frac{1}{4} \).Trigonometric values like these are common in calculus due to their known and specific behavior at standard angle values. Handling trigonometric limits is a fundamental skill that surfaces in many calculus scenarios.

Vector components

Vector components are individual segments that make up a vector in mathematics. Each component represents a different aspect of the vector, behaving independently but contributing to the overall vector's outcome.
In vector-valued functions, evaluating limits or any operation means treating each part as a separate entity before putting them back together.

Take our problem, where:

• The vector \( \langle \cos^2 t, \sin^2 t, 1 \rangle \) is split into its three components.
• Both trigonometric calculations for cosine and sine are performed individually.
• The constant \(1\) stays unchanged since its limit is itself.

Upon finding the limit of each component, they are combined back to form \[ \langle \frac{3}{4}, \frac{1}{4}, 1 \rangle \]Understanding vector components aids in visualizing how each part influences the behavior of the entire vector. It's essential to grasp how changes in one direction do not directly affect the other, showcasing the independent yet collective identity of vector components.