Question

Find the required limit or indicate that it does not exist. $$ \lim _{t \rightarrow 0^{-}}\left\langle e^{-1 / t^{2}}, \frac{t}{|t|},|t|\right\rangle $$

Short answer

The limit is \( \langle 0, -1, 0 \rangle \).

Step-by-step solution

Analyze the vector components

The given vector function is \( \lim _{t \rightarrow 0^{-}}\langle e^{-1 / t^{2}}, \frac{t}{|t|}, |t| \rangle \). We need to evaluate the limit for each component separately. The components are \( e^{-1/t^2} \), \( \frac{t}{|t|} \), and \( |t| \).

Evaluate the first component

Consider the first component, \( e^{-1/t^2} \). As \( t \rightarrow 0^{-} \), \(-1/t^2\) goes to \(-\infty\), so the expression \( e^{-1/t^2} \to 0\). Thus, the limit of the first component is 0.

Evaluate the second component

Now, look at the second component, \( \frac{t}{|t|} \). For \( t \rightarrow 0^{-} \), \( t \) is negative. Hence \( |t| = -t \), and \( \frac{t}{|t|} = \frac{t}{-t} = -1 \). The limit of the second component is \(-1\).

Evaluate the third component

For the third component, \( |t| \), as \( t \rightarrow 0^{-} \), \( t \) is negative and \(|t| = -t\). Therefore, \( |t| \rightarrow 0\). The limit for the third component is 0.

Combine the results

The limit of the vector \( \langle e^{-1/t^2}, \frac{t}{|t|}, |t| \rangle \) as \( t \rightarrow 0^{-} \) is the vector formed by the limits of each component: \( \langle 0, -1, 0 \rangle \).

Key concepts

Vector Functions

Vector functions are a fundamental concept in calculus and physics, offering a way to describe curves and paths in space. These functions map points from one space to another, often from a single variable, like time, to a multi-dimensional space.

For example, consider a vector function expressed as \( \mathbf{r}(t) = \langle f(t), g(t), h(t) \rangle \), where \( t \) is a parameter. Here, a change in \( t \) results in changes to each component of the vector, tracing a curve in three-dimensional space.

• The vector function can be visualized in a coordinate system, with each component representing a dimension (e.g., x, y, z).
• These functions are useful for modeling real-world situations, such as the path of an object, changes in velocity, or the gradient of a physical field.
• They provide insight into how different variables interact over time or space.

Evaluating limits of vector functions involves understanding each component separately, enabling an understanding of the overall behavior of the curve or path as certain parameters change.

Component-Wise Limits

To evaluate the limit of a vector function, we often use the component-wise approach. This involves analyzing each component of the vector function independently and then synthesizing the results.

Consider a vector function \( \mathbf{v}(t) = \langle x(t), y(t), z(t) \rangle \). To find \( \lim_{t \to a} \mathbf{v}(t) \), we find the limit of each individual component:

• \( \lim_{t \to a} x(t) \)
• \( \lim_{t \to a} y(t) \)
• \( \lim_{t \to a} z(t) \)

Once these limits are determined, the limit of the vector function is \( \langle \lim_{t \to a} x(t), \lim_{t \to a} y(t), \lim_{t \to a} z(t) \rangle \).

By studying each part separately, this approach clarifies how each component behaves as \( t \) approaches a specific value, making it more manageable to understand complex vector functions and their limits. This method ensures that each factor is considered to create a complete picture of the vector function's behavior at that point.

One-Sided Limits

One-sided limits are a handy tool for understanding the behavior of functions as they approach a specific point from one side—either from the left or the right.

Specifically, for a vector function \( \langle f(t), g(t), h(t) \rangle \), as \( t \) approaches a vector's component value from one side, we describe:

• The left-hand limit (notation: \( \lim_{t \to a^-} \)) when approaching from values less than \( a \).
• The right-hand limit (notation: \( \lim_{t \to a^+} \)) when approaching from values more than \( a \).

In our exercise, we evaluated the left-hand limit where \( t \rightarrow 0^{-} \) for each component.

This concept is central to studying discontinuities, assuring whether limits exist as a function approaches a boundary from one side alone, adding clarity to the overall behavior of vector functions in specific situations.

Exponential Decay

Exponential decay describes a process where quantities decrease at rates proportional to their current value, a concept showcased through functions like \( e^{-x} \). As \( x \to \infty \), exponential terms approach zero, reflecting rapid decline.

This phenomenon is common in nature and physics, applicable to processes such as radioactive decay, heat dissipation, and more.

• The expression \( e^{-1/t^2} \) is an example of exponential decay, where as \( t \to 0^- \), \(-\frac{1}{t^2} \) increases towards negative infinity.
• Therefore, \( e^{-1/t^2} \to 0 \), indicating an exceedingly fast decay rate to zero, emphasizing the swift reduction typical of exponential functions.
• Learning about this type of decay helps in modeling scenarios where a swift decline happens, providing insight into solutions for natural decay processes and financial devaluation.

Understanding exponential decay's impact on components of vector functions can enrich insight into more complex multi-dimensional changes.