Problem 45 - 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 if $$0<|t-a|<\delta\( then \)|\mathbf{r}(t)-\mathbf{b}|<\varepsilon$$

Step-by-Step Solution

Answer

The limit definition is satisfied: \\( \lim_{t \rightarrow a} \mathbf{r}(t) = \mathbf{b} \\\) if and only if for every \\( \varepsilon > 0 \\\), there exists \\( \delta > 0 \\\) such that \\( 0<|t-a|<\delta \\\) implies \\( |\mathbf{r}(t)-\mathbf{b}|<\varepsilon \\\).

Step 1: Understand the Limit Definition

The limit definition states that the limit of a function \( \mathbf{r}(t) \) as \( t \) approaches \( a \) is equal to \( \mathbf{b} \) if for every \( \varepsilon > 0 \), there exists a corresponding \( \delta > 0 \) such that whenever \( 0 < |t - a| < \delta \), it follows that \( |\mathbf{r}(t) - \mathbf{b}| < \varepsilon \). We need to prove this bidirectional implication.

Step 2: Prove the 'If' Direction

Assume that \( \lim_{t \rightarrow a} \mathbf{r}(t) = \mathbf{b} \). By definition, for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) such that if \( 0 < |t-a| < \delta \), then the condition \( |\mathbf{r}(t) - \mathbf{b}| < \varepsilon \) is satisfied. This directly fulfills the given condition of the limit definition, confirming the 'if' portion of the statement.

Step 3: Prove the 'Only If' Direction

Next, assume that for every \( \varepsilon > 0 \), there exists a \( \delta > 0 \) ensuring \( 0 < |t-a| < \delta \) implies \( |\mathbf{r}(t) - \mathbf{b}| < \varepsilon \). This matches the condition needed for \( \lim_{t \rightarrow a} \mathbf{r}(t) = \mathbf{b} \) according to the standard \( \varepsilon-\delta \) definition of limits. Therefore, this assumption implies that \( \lim_{t \rightarrow a} \mathbf{r}(t) = \mathbf{b} \), proving the 'only if' portion.

Step 4: Conclusion

Both directions have been proved: 1) Assuming the limit exists leads to the \( \varepsilon-\delta \) condition, and 2) Assuming the \( \varepsilon-\delta \) condition leads to the existence of the limit. Thus, this is an 'if and only if' statement, establishing that the conditions are equivalent as required.

Key Concepts

Understanding Limits in Calculus

A limit in calculus is a fundamental concept that describes the behavior of a function as its input approaches a specific value. In simple terms, when we talk about \( \lim_{t \to a} \mathbf{r}(t) = \mathbf{b} \), we mean that as \( t \) gets closer to \( a \), the value of the function \( \mathbf{r}(t) \) gets closer to \( \mathbf{b} \). This concept of limits allows mathematicians to understand how functions behave near specific points and is essential for defining concepts like derivatives and integrals.

Some key ideas to remember about limits include:

• Limits can be approached from either direction of a point \( a \). These are called one-sided limits.
• A limit exists if the function approaches the same value no matter the side from which \( t \) approaches \( a \).
• The \( \varepsilon-\delta \) definition rigorously verifies the limit's existence through the concept of distance and closeness.

Understanding limits is crucial for tackling more complex topics in calculus, including continuity and differentiability. Limits help provide a formal framework within which these more advanced ideas can be rigorously explored.

Exploring Vector-Valued Functions

Vector-valued functions are an intriguing concept in calculus where the output is not a single real number, but a vector. When we look at functions like \( \mathbf{r}(t) \), we are interested in how the components of the vector change with respect to the input variable \( t \). These functions are critical in fields like physics and engineering, where objects have multiple degrees of freedom.

Some features of vector-valued functions include:

• These functions take a scalar input and return a vector, often represented as \( \mathbf{r}(t) = \langle r_1(t), r_2(t), \, r_3(t) \rangle \).
• Each component function (\( r_1(t), r_2(t), \text{etc.} \)) is a real-valued function itself.
• Limits and continuity can be examined component-wise, reducing the complexity of analysis.

Vector-valued functions become indispensable tools in analyzing motion along curves in space, electromagnetic fields, and various engineering dynamics. They expand the reach of traditional calculus to multi-dimensional spaces and significantly enrich mathematical modeling capability.

Conceptualizing Continuity

Continuity is a fundamental concept that follows closely from limits. In simple words, a function is continuous at a point \( a \) if the limit of the function as \( t \) approaches \( a \) is the same as the function's value at \( a \), i.e., \( \lim_{t \to a} \mathbf{r}(t) = \mathbf{r}(a) \).

To grasp the idea of continuity:

• A continuous function has no jumps, breaks, or holes at the point \( a \).
• Continuity can be examined through the epsilon-delta definition, where for a function \( f \) to be continuous at \( a \), for every \( \varepsilon > 0 \), there must be a \( \delta > 0 \) such that whenever \( |t-a| < \delta \), \(|f(t) - f(a)| < \varepsilon \).
• Continuity across an interval means the function is continuous at every point within that interval.

The idea of continuity is key for ensuring that functions behave predictably. It's a central concept in analysis and is essential for defining other properties like differentiability. In the context of vector-valued functions, continuity is checked for each component function, assuring a smooth path through space.