Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

In Problems \(1-8\), find the required limit or indicate that it does not exist. $$ \lim _{t \rightarrow 1}\left[2 t \mathbf{i}-t^{2} \mathbf{j}\right] $$

Step-by-Step Solution

Answer

The limit is \( 2\mathbf{i} - \mathbf{j} \).

Step 1: Recognize the Vector Function Components

The given problem involves finding the limit of a vector function as \( t \) approaches 1. The vector function is \( 2t \mathbf{i} - t^2 \mathbf{j} \), where \( 2t \) is the \( i \)-component and \( -t^2 \) is the \( j \)-component.

Step 2: Find the Limit of Each Component Separately

To find the limit of the vector, we must find the limit of each of its components as \( t \rightarrow 1 \). The limit of the \( i \)-component is \( \lim_{t \to 1} 2t \) and the limit of the \( j \)-component is \( \lim_{t \to 1} (-t^2) \).

Step 3: Calculate the Limit of the i-Component

Evaluate \( \lim_{t \to 1} 2t \). Substituting \( t = 1 \) into \( 2t \), we get \( 2 \times 1 = 2 \). Thus, the limit of the \( i \)-component is 2.

Step 4: Calculate the Limit of the j-Component

Evaluate \( \lim_{t \to 1} (-t^2) \). Substituting \( t = 1 \) into \( -t^2 \), we get \(-1^2 = -1\). Thus, the limit of the \( j \)-component is -1.

Step 5: Combine the Limits to Find the Vector Limit

Combine the results of the limits of the components into a vector form: \( 2\mathbf{i} - 1\mathbf{j} \). Therefore, the limit of the vector function is \( 2\mathbf{i} - \mathbf{j} \).

Key Concepts

Vector Functions

A vector function is an essential concept in calculus and vector calculus. It involves functions that return vectors instead of scalar values. When dealing with vector functions, each output at a given point is a vector composed of multiple components. For instance, the vector function in our exercise is given as \( 2t \mathbf{i} - t^2 \mathbf{j} \). Each part of this vector function, \( 2t \) and \( -t^2 \), represents different components of the vector, specifically along the \( \mathbf{i} \) and \( \mathbf{j} \) directions respectively.
Vector functions are useful because they can describe movements and forces in multiple dimensions, such as a curve in 3D space. When learning about vector functions, it's important to understand how they relate to their component parts, providing both magnitude and direction for each unit vector involved.

Component-Wise Limits

A helpful approach when evaluating the limits of vector functions is the component-wise method. This means you can break down a vector into its component functions and consider each one separately. For example, in order to find the limit of the vector function \( 2t \mathbf{i} - t^2 \mathbf{j} \) as \( t \) approaches 1, we analyze the individual limits:

• \( \lim_{t \to 1} 2t \)
• \( \lim_{t \to 1} (-t^2) \)

By evaluating each component separately, the calculation becomes more straightforward. You determine the behavior of each part as \( t \) approaches a certain value and then recombine them into a final vector form.
This method highlights the simplicity of analyzing vector functions by treating them as collections of simpler scalar functions.

Calculus Limits

Limits play a central role in calculus, providing a way to understand the behavior of functions as inputs approach certain values. The concept of a limit helps in investigating function continuity, derivatives, and integrals. Limits allow us to examine how functions behave near a point without necessarily having to compute the function's value exactly at that point.
When working with limits, it is crucial to substitute the approaching value into the function whenever possible, to observe the tendency. For the vector function in question, once you substitute \( t = 1 \) into the respective components \( 2t \) and \( -t^2 \), you get specific concrete values without ambiguity. This step ensures the calculation of the individual component limits to be merged into the overall limit of the vector.

Vector Calculus

Vector calculus extends regular calculus into multiple dimensions, dealing with vector fields and operations like gradients, curls, and divergences. Understanding vector limits forms the foundational knowledge needed to explore advanced topics within vector calculus. In the realm of vector calculus, analyzing how vector functions behave as they approach certain points is essential for many applications, such as fluid dynamics, electromagnetism, and other fields involving vector quantities.
By practicing limits of vector functions, you develop an intuition for how components of vectors interact and change. These exercises lay the groundwork for tackling more complex vector calculus problems where analyzing directional derivatives and multi-dimensional surfaces is necessary.
Mastering concepts like vector functions, component-wise operations, and calculus limits in the context of vector calculus paves the way for successful applications in physics and engineering.