Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \lim _{t \rightarrow 0}\left[\frac{\sin t \cos t}{t} \mathbf{i}-\frac{7 t^{3}}{e^{t}} \mathbf{j}+\frac{t}{t+1} \mathbf{k}\right] $$

Step-by-Step Solution

Answer

The limit is \( \mathbf{i} \).

Step 1: Separate Vector Components

The given limit involves three components for vector **i**, **j**, and **k**. Each component will be handled separately.- For the **i** component: \( \lim_{t \to 0} \left( \frac{\sin t \cos t}{t} \right) \)- For the **j** component: \( \lim_{t \to 0} \left( -\frac{7t^3}{e^t} \right) \)- For the **k** component: \( \lim_{t \to 0} \left( \frac{t}{t+1} \right) \)

Step 2: Compute the Limit for i Component

For \( \lim_{t \to 0} \left( \frac{\sin t \cos t}{t} \right) \), apply the small angle approximations \( \sin t \approx t \) and \( \cos t \approx 1 \) when \( t \to 0 \). Thus, the expression becomes:\[ \lim_{t \to 0} \left( \frac{\sin t \cos t}{t} \right) = \lim_{t \to 0} \frac{t \cdot 1}{t} = \lim_{t \to 0} 1 = 1 \]So, the limit for the **i** component is 1.

Step 3: Compute the Limit for j Component

For \( \lim_{t \to 0} \left( -\frac{7t^3}{e^t} \right) \), notice that both \( 7t^3 \to 0 \) and \( e^t \to 1 \) when \( t \to 0 \). Thus:\[ \lim_{t \to 0} \left( -\frac{7t^3}{e^t} \right) = \lim_{t \to 0} \left( -\frac{0}{1} \right) = 0 \]Thus, the limit for the **j** component is 0.

Step 4: Compute the Limit for k Component

For \( \lim_{t \to 0} \left( \frac{t}{t+1} \right) \), plug in \( t = 0 \):\[ \lim_{t \to 0} \frac{t}{t+1} = \frac{0}{1} = 0 \]Therefore, the limit for the **k** component is 0.

Step 5: Combine the Component Limits

Combine the computed limits for each component from previous steps to find the vector limit:\[ \lim_{t \to 0} \left[ \frac{\sin t \cos t}{t} \mathbf{i} - \frac{7t^3}{e^t} \mathbf{j} + \frac{t}{t+1} \mathbf{k} \right] = 1 \mathbf{i} + 0 \mathbf{j} + 0 \mathbf{k} = \mathbf{i} \]

Key Concepts

Limit of a Vector Function

Understanding limits of vector functions involves evaluating the behavior of each component of the vector as the variable approaches a certain value. In the given problem, the vector function is parameterized by the variable \( t \) as it approaches 0. Each component of the vector must be evaluated separately to understand how the vector function behaves in the limit.A vector function can be thought of as a set of individual functions for each standard unit vector component \( \mathbf{i}, \mathbf{j}, \mathbf{k} \), etc. The limit of a vector function is the vector composed of the limits of its components. This approach is essential in multivariable calculus where vector fields and vector-valued functions appear frequently.

Component-wise Limit Evaluation

To find the limit of a vector function, evaluating each component separately is crucial. This method is called component-wise limit evaluation. It helps in simplifying complex expressions by focusing on smaller, more manageable parts.

• For the \( \mathbf{i} \) component, the limit was calculated by simplifying \( \lim_{t \to 0} \left( \frac{\sin t \cos t}{t} \right) \), which results in 1.
• The \( \mathbf{j} \) component, \( \lim_{t \to 0} \left( -\frac{7t^3}{e^t} \right) \), simplifies to 0 as both the numerator goes to 0 faster than the exponential term.
• Finally, for the \( \mathbf{k} \) component, substituting directly \( \lim_{t \to 0} \left( \frac{t}{t+1} \right) \) gives 0.

Each component converges towards its limit independently, allowing the vector limit to be reassembled in the end.

Small Angle Approximations

When finding limits involving trigonometric functions, like \( \sin t \) and \( \cos t \), small angle approximations can be invaluable. As \( t \) approaches 0, some useful approximations are:

• \( \sin t \approx t \)
• \( \cos t \approx 1 \)

These approximations simplify complex expressions, making it easier to calculate limits or differentiate these functions.For example, in the \( \mathbf{i} \) component, \( \frac{\sin t \cos t}{t} \) was simplified using these approximations, reducing the expression to 1. This is crucial for understanding behaviors near zero and is often used in physics and engineering to simplify small angle problems.

Vector Algebra in Calculus

Vector algebra plays a crucial role in calculus, especially in multivariable calculus, where vectors and operations on them are quite common.Vector functions often require combining results from multiple separate parts, akin to the process seen in breaking down components with the \( \mathbf{i}, \mathbf{j}, \mathbf{k} \) axes. After evaluating each component's limit, they are combined back to get the final result. In this case, the vector has the limit \( \mathbf{i} \), which means the vector aligns with the x-axis at \( t \to 0 \).This methodical approach of component separation and combination ensures accuracy, making calculations more structured. Understanding how vector algebra integrates with calculus opens doors to analyzing systems where direction and magnitude are essential, like physics-related vector fields, ensuring a comprehensive grasp of how vectors behave under various transformations.