Problem 15 - Advanced Engineering Mathematics

Question

Given that \(\mathbf{r}(t)=\frac{\sin \angle t}{t} \mathbf{i}+(t-2)^{5} \mathbf{j}+t \ln t \mathbf{k}\), find \(\lim _{t \rightarrow 0^{+}} \mathbf{r}(t)\).

Step-by-Step Solution

Answer

The limit is \( \mathbf{i} - 32 \mathbf{j} \).

Step 1: Determine Each Component

The vector function is given as \( \mathbf{r}(t) = \frac{\sin t}{t} \mathbf{i} + (t-2)^{5} \mathbf{j} + t \ln t \mathbf{k} \). We need to find the limit for each component separately as \( t \to 0^{+} \).

Step 2: Evaluate Limit of First Component

Consider \( \frac{\sin t}{t} \). As \( t \to 0 \), \( \frac{\sin t}{t} \to 1 \). This is a standard limit result.

Step 3: Evaluate Limit of Second Component

Consider \( (t-2)^{5} \). As \( t \to 0 \), the expression \( t-2 \to -2 \), hence \( (t-2)^{5} \to (-2)^{5} = -32 \).

Step 4: Evaluate Limit of Third Component

Consider \( t \ln t \). As \( t \to 0^{+} \), we transform this using L'Hôpital's Rule by rewriting as \[ \lim_{t \to 0^{+}} \frac{\ln t}{1/t} \]. Apply L'Hôpital's Rule: differentiate numerator and denominator to get \[ \lim_{t \to 0^{+}} \frac{1/t}{-1/t^2} = \lim_{t \to 0^{+}} -t = 0 \].

Step 5: Assemble the Limits

Combine the evaluated limits for each component to form the limit of the vector function: \( \lim_{t \to 0^{+}} \mathbf{r}(t) = 1 \mathbf{i} - 32 \mathbf{j} + 0 \mathbf{k} \). Thus, the result is \( \mathbf{i} - 32 \mathbf{j} \).

Key Concepts

Limit of Vector Functions

Understanding the limit of vector functions involves calculating the limit of each component of the vector separately. A vector function is generally defined in terms of its i, j, and k components, each of which can be a function of a parameter, like time or space.

• For the vector function \( \mathbf{r}(t) = \frac{\sin t}{t} \mathbf{i} + (t-2)^{5} \mathbf{j} + t \ln t \mathbf{k} \), we evaluate the limits of \( \frac{\sin t}{t} \), \( (t-2)^5 \), and \( t \ln t \) separately as \( t \to 0^{+} \).
• The approach of evaluating each component's limit is critical because each function within the vector might have a different behavior as the parameter approaches a specific value.
• This means that even when handling a complex vector function, we break it down into simpler, more manageable parts.

The limiting process simplifies the analysis and helps in understanding complex systems and their responses in areas like physics or engineering. Once the components' limits are calculated, they are combined to find the overall limit of the vector function.

L'Hôpital's Rule

L'Hôpital's Rule is a powerful tool in calculus used to evaluate limits involving indeterminate forms, such as \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \). The rule states that for functions \( f(t) \) and \( g(t) \) which are differentiable,

• if \( \lim_{t \to a} \frac{f(t)}{g(t)} \) yields an indeterminate form, then \( \lim_{t \to a} \frac{f(t)}{g(t)} = \lim_{t \to a} \frac{f'(t)}{g'(t)} \) provided the limit on the right-hand side exists.
• In our exercise, to find the limit of \( t \ln t \) as \( t \to 0^{+} \), we rewrote this as a fraction: \( \lim_{t \to 0^{+}} \frac{\ln t}{1/t} \).
• Upon differentiating, we found \( \frac{1/t}{-1/t^2} \) yielding \( -t \). Thus, the limit simplifies to \( 0 \) as \( t \to 0^{+} \).

L'Hôpital's Rule simplifies complex limits into more manageable expressions, making it an invaluable technique for solving calculus problems involving vector functions or any intricate expressions.

Vector Components

Vectors are essential in representing quantities with both magnitude and direction. A vector in a 3D space is typically expressed in terms of its components along the coordinate axes:

• The \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) represent the unit vectors along the x, y, and z axes, respectively.
• When a vector is given by \( \mathbf{r}(t) = f(t) \mathbf{i} + g(t) \mathbf{j} + h(t) \mathbf{k} \), each function \( f(t) \), \( g(t) \), and \( h(t) \) defines how the vector behaves in its respective direction.
• Breaking down vectors into components provides a clearer understanding of their effect in each independent direction and simplifies mathematical operations such as addition, subtraction, and finding limits.

Understanding vector components is crucial because it allows us to manage complex systems by focusing on one direction at a time. We accurately capture the essence of motion or forces in terms of simple functions, enabling easier computation and interpretation.