Problem 11 - Calculus III

Question

Find the limit of the following vector-valued functions at the indicated value of \(t\). $$ \lim _{t \rightarrow e^{2}}\left\langle t \ln (t), \frac{\ln t}{t^{2}}, \sqrt{\ln \left(t^{2}\right)}\right\rangle $$

Step-by-Step Solution

Answer

The limit is \(\left\langle 2e^2, \frac{2}{e^4}, 2 \right\rangle\)."

Step 1: Understand the Problem

We need to find the limit of the vector-valued function \(\displaystyle \vec{F}(t) = \left\langle t \ln (t), \frac{\ln t}{t^{2}}, \sqrt{\ln \left(t^{2}\right)}\right\rangle\) as \(t\) approaches \(e^2\). This involves computing the limit of each component separately.

Step 2: Limit of First Component

Consider the first component, \(f(t) = t\ln(t)\). Substitute \(t = e^2\) into this function: \[ \lim_{t \to e^2} t\ln(t) = e^2 \ln(e^2) = e^2 \cdot 2 = 2e^2. \] Thus, the limit of the first component is \(2e^2\).

Step 3: Limit of Second Component

For the second component, \(g(t) = \frac{\ln(t)}{t^2}\). Evaluate the limit as \(t\) approaches \(e^2\): \[ \lim_{t \to e^2} \frac{\ln(t)}{t^2} = \frac{\ln(e^2)}{(e^2)^2} = \frac{2}{e^4}. \] Thus, the limit of the second component is \(\frac{2}{e^4}\).

Step 4: Limit of Third Component

Examine the third component, \(h(t) = \sqrt{\ln(t^2)}\). Recognize that \(\ln(t^2) = 2\ln(t)\). The limit is: \[ \lim_{t \to e^2} \sqrt{\ln(t^2)} = \sqrt{2\ln(e^2)} = \sqrt{4} = 2. \] Therefore, the limit of the third component is 2.

Step 5: Combine Component Limits

Combine the limits of all components to find the limit of the vector-valued function: \[ \lim _{t \rightarrow e^{2}}\left\langle t \ln (t), \frac{\ln t}{t^{2}}, \sqrt{\ln \left(t^{2}\right)}\right\rangle = \left\langle 2e^2, \frac{2}{e^4}, 2 \right\rangle.\] This is the final result for the limit of the function as \(t\) approaches \(e^2\).

Key Concepts

Limits

Limits are a fundamental concept in calculus. They help us understand the behavior of functions as they approach a particular point. Think of a limit as the value that a function is approaching, even if it never quite reaches that value directly.
In the context of vector-valued functions, which have multiple components, we compute the limit of each component separately. This means we break down the vector into its individual parts and determine the limit of each part independently.

• This approach simplifies the problem because we're essentially dealing with simpler, one-dimensional limits.
• Once we compute limits for all components, we can combine them to find the overall limit of the vector-valued function.

Understanding limits is crucial as they lay the groundwork for further topics in calculus, like derivatives and integrals.

Calculus III

Calculus III, often viewed as the next step in the calculus sequence, builds on the principles learned in Calculus I and II. It delves deeply into multivariable calculus, expanding our understanding of functions and limits.
This includes working with functions that have more than one variable. Vector-valued functions, as seen in the problem, are an integral part of Calculus III.

• In these courses, students explore concepts like partial derivatives, multiple integrals, and vectors.
• The focus shifts from examining curves in one-dimensional space to curves and surfaces in multi-dimensional space.

Mastering these concepts requires a good grasp of vector operations and the ability to visualize mathematical problems in multiple dimensions.

Component-Wise Limit

The concept of a component-wise limit is crucial when dealing with vector-valued functions. It refers to the method of finding the limit of a vector function by evaluating the limit of each of its components separately.
This approach simplifies the process because you apply familiar limit techniques to individual components, making complex vector functions more manageable.

• Consider each component function as a separate, smaller problem.
• Calculate the limit of each one using standard methods, such as factoring, substitution, or L'Hôpital's rule, if applicable.

Once you have the limits of the individual components, you combine them to find the limit of the entire vector function. This step-by-step strategy helps break down complicated problems into easier, solvable parts.