Problem 13 - Calculus III

Question

Find the limit of the following vector-valued functions at the indicated value of \(t\). $$ \lim _{t \rightarrow \infty} \mathbf{r}(t) \text { for } \mathbf{r}(t)=2 e^{-t} \mathbf{i}+e^{-t} \mathbf{j}+\ln (t-1) \mathbf{k} $$

Step-by-Step Solution

Answer

The vector-valued function approaches \( \infty \mathbf{k} \) as \( t \rightarrow \infty \).

Step 1: Identify the Components

The vector-valued function is given as \( \mathbf{r}(t) = 2e^{-t} \mathbf{i} + e^{-t} \mathbf{j} + \ln(t-1) \mathbf{k} \). We need to evaluate the limit of each component individually as \( t \rightarrow \infty \).

Step 2: Find Limit of the First Component

Consider the first component, \( 2e^{-t} \). As \( t \rightarrow \infty \), \( e^{-t} \rightarrow 0 \), so multiplying by 2 still results in \( 2 imes 0 = 0 \). Thus, \( \lim_{t \to \infty} 2e^{-t} = 0 \).

Step 3: Find Limit of the Second Component

For the second component, \( e^{-t} \), similarly, as \( t \rightarrow \infty \), \( e^{-t} \rightarrow 0 \). Therefore, \( \lim_{t \to \infty} e^{-t} = 0 \).

Step 4: Find Limit of the Third Component

For the third component, \( \ln(t-1) \), as \( t \rightarrow \infty \), \( \ln(t-1) \rightarrow \infty \). So, the limit is \( \lim_{t \to \infty} \ln(t-1) = \infty \).

Step 5: Combine the Limits of the Components

Now, combine the evaluated limits for each component. The limit of the vector-valued function is \( \lim_{t \to \infty} \mathbf{r}(t) = 0 \mathbf{i} + 0 \mathbf{j} + \infty \mathbf{k} \). This simplifies to \( \mathbf{r}(t) \rightarrow \infty \mathbf{k} \).

Key Concepts

Understanding Limits in Vector-Valued Functions

When dealing with vector-valued functions, to find the limit as a parameter approaches a certain value, we evaluate each component of the vector independently. In our exercise, the vector-valued function is composed of three components:

• \(2e^{-t}\) for the \(\mathbf{i}\) component
• \(e^{-t}\) for the \(\mathbf{j}\) component
• \(\ln(t-1)\) for the \(\mathbf{k}\) component

As \(t\) approaches infinity, we solve for the limit of each component separately.

When handling limits, it's crucial to understand that functions involving exponential decay (like \(e^{-t}\)) approach zero as \(t\) increases.
In contrast, natural logarithms like \(\ln(t-1)\) increase towards infinity as \(t\) approaches infinity.
By interpreting each component's behavior independently, we can combine their limits to determine the overall limit of the vector function.

Exponential Functions and Their Limits

Exponential functions are incredibly powerful in calculus due to their unique properties.
An exponential function of the form \(e^{-t}\) is characterized by a base \(e\), Euler's number, which is approximately 2.71828.
This function depicts exponential decay. For \(e^{-t}\), as the exponent \(-t\) becomes very large (in the negative direction), the value of the function rapidly approaches zero.

In our exercise, both \(2e^{-t}\) and \(e^{-t}\) are exponential functions.
The component \(2e^{-t}\) scales this decay by a factor of 2, yet, as \(t\) trends toward infinity, the overall value still zeroes out.

• The fact \(\lim_{t \to \infty} e^{-t} = 0\) highlights the behavior of exponential functions shrinking towards zero in the presence of negative growth.
• This shrinking also means, regardless of any multiplicative constants, the dominant factor remains, leading to simplicity in limit calculations for such expressions.

Recognizing the effects of exponential decay in vector-valued functions aids in predicting their long-term behavior in various applications.

Diving into the Natural Logarithm

The natural logarithm, denoted \(\ln(x)\), is the inverse function of the exponential function \(e^x\).
It is used to determine the time needed for a particular growth rate to occur given a specific amount.

In contexts where \(t\) approaches infinity, \(\ln(t-1)\) can be thought of as measuring the time required to reach certain growth.
Not surprisingly, as \(t\) increases endlessly, \(\ln(t-1)\) also grows without bound, leading towards infinity.

• The function automates processes such as data scaling and growth measurement.
• In physics and engineering contexts, the natural logarithm captures behavior related to the time or scale factor in processes.

In our vector function example, this property of \(\ln(t-1)\) results in the overarching trend of direction towards positive infinity along the \(\mathbf{k}\) axis when combining component limits.