Problem 70 - Essential Calculus Early Transcendentals

Question

Show that \(\lim _{t \rightarrow a} \mathbf{r}(t)=\mathbf{b}\) if and only if for every \(\varepsilon>0\) there is a number \(\delta>0\) such that \(|\mathbf{r}(t)-\mathbf{b}|<\varepsilon\) whenever \(0<|t-a|<\delta\)

Step-by-Step Solution

Answer

The statement and limit definition are equivalent by showing both directions: if limit holds, condition holds, and vice versa.

Step 1: Understand the Limit Definition

The question asks to show the definition of a limit for vectors. The vector function \( \mathbf{r}(t) \) approaches the vector \( \mathbf{b} \) as \( t \) approaches \( a \). For this to happen, for every positive number \( \varepsilon \), there must exist a positive number \( \delta \) such that whenever \( 0 < |t-a| < \delta \), the vector distance \( |\mathbf{r}(t) - \mathbf{b}| \) is less than \( \varepsilon \).

Step 2: Assume Limit Exists to Prove Forward Implication

Assume \( \lim_{t \rightarrow a} \mathbf{r}(t)=\mathbf{b} \). This implies that as \( t \) gets arbitrarily close to \( a \), \( \mathbf{r}(t) \) gets arbitrarily close to \( \mathbf{b} \). By definition, for every \( \varepsilon > 0 \), there must exist a \( \delta > 0 \) such that whenever \( 0 < |t-a| < \delta \), \( |\mathbf{r}(t) - \mathbf{b}| < \varepsilon \). Thus, the forward implication is established.

Step 3: Prove Reverse Implication

Conversely, assume for every \( \varepsilon > 0 \), a \( \delta > 0 \) exists such that \( 0 < |t-a| < \delta \) implies \( |\mathbf{r}(t) - \mathbf{b}| < \varepsilon \). This validates that as \( t \) approaches \( a \), \( \mathbf{r}(t) \) approaches \( \mathbf{b} \). Therefore, \( \lim_{t \rightarrow a} \mathbf{r}(t) = \mathbf{b} \). Thus, the reverse implication is also shown.

Key Concepts

Vector Functions

In calculus, functions of vectors or **vector functions** are often used to represent objects that change over time or space, such as the position of a moving particle. Instead of returning a single real number, vector functions return vectors. These vectors can have two or three components, representing dimensions in space, but they can be higher-dimensional as well.

A typical representation of a vector function is \( \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle \), where \( f(t), g(t), \) and \( h(t) \) are real-valued functions of \( t \). Here, **\( t \)** is often thought of as time, and **\( \mathbf{r}(t) \)** describes the trajectory of a particle in a plane or space.

• The vector function maps a real number \( t \) to a vector in a multidimensional space.
• Each component function \( f(t), g(t), h(t) \) changes with \( t \), allowing us to visualize movement.

Understanding vector functions is crucial for analyzing trajectories and behaviors across multiple dimensions.

Epsilon-Delta Definition

The **epsilon-delta definition** is a foundational concept in calculus, used to define what it means for a function's output to "approach" a particular value as its input approaches a specific point. For vector functions, the definition is extended to vector outputs.

Using the problem statement, the epsilon-delta definition for a vector limit says: for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |t-a| < \delta \), then \( |\mathbf{r}(t) - \mathbf{b}| < \varepsilon \).

• \( \varepsilon \) represents how close \( \mathbf{r}(t) \) needs to be to \( \mathbf{b} \).
• \( \delta \) shows how close \( t \) must be to \( a \) for this closeness in function value.

This definition ensures the accuracy of approaching the limit, framing the relationship between the inputs, outputs, and desired proximity to a target value.

Calculus Limit Proofs

In mathematics, proving limits, especially in the context of **calculus limit proofs**, involves demonstrating that a function behaves predictably as its input approaches a particular point. For vector functions, proof strategies are similar to those for scalar functions, but they accommodate the vector nature of outputs.

The proof involves two steps:

• **Forward Implication**: Assume \( \lim_{t \rightarrow a} \mathbf{r}(t) = \mathbf{b} \) and show for \( \varepsilon > 0 \) that \( \delta > 0 \) exists, meeting the condition \( |\mathbf{r}(t) - \mathbf{b}| < \varepsilon \) if \( 0 < |t-a| < \delta \).
• **Reverse Implication**: Start with the assumption that for every \( \varepsilon > 0 \), a \( \delta > 0 \) exists such that \( |\mathbf{r}(t) - \mathbf{b}| < \varepsilon \) when \( 0 < |t-a| < \delta \), leading to the conclusion that \( \lim_{t \rightarrow a} \mathbf{r}(t) = \mathbf{b} \).

These steps solidify the understanding of vector limit behavior and validate the limit's existence, crucial for deeper explorations in multivariable calculus.