Problem 8 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \lim _{t \rightarrow 0^{-}}\left\langle e^{-1 / t^{2}}, \frac{t}{|t|},|t|\right\rangle $$

Step-by-Step Solution

Answer

\( \langle 0, -1, 0 \rangle \)

Step 1: Analyze Each Component Separately

We begin by analyzing the components of the vector as \( t \to 0^- \) . The first component is \( e^{-1/t^2} \). As \( t \) approaches 0 from the negative side, \( 1/t^2 \) becomes very large, so \( e^{-1/t^2} \to e^{- ext{large number}} \to 0 \).

Step 2: Analyze the Second Component

The second component is \( \frac{t}{|t|} \). As \( t \to 0^- \), \( t < 0 \) and \( |t| = -t \). Thus, \( \frac{t}{|t|} = \frac{t}{-t} = -1 \).

Step 3: Analyze the Third Component

The third component is \( |t| \), which is simply the absolute value of \( t \). As \( t \to 0^- \), \( t < 0 \) implies that \( |t| = -t \). As \( t \to 0^- \), \( -t \to 0^+ \). Therefore, the limit of \( |t| \) as \( t \to 0^- \) is 0.

Step 4: Combine the Limits of All Components

Having analyzed each component separately, we can combine them to evaluate the limit of the vector as \( t \to 0^- \):\[ \lim _{t ightarrow 0^{-}} \left\langle e^{-1 / t^{2}}, \frac{t}{|t|}, |t| \right\rangle = \left\langle 0, -1, 0 \right\rangle.\]

Key Concepts

Multivariable Calculus

Multivariable calculus is an extension of calculus that deals with functions of more than one variable. It allows us to explore mathematical phenomena in higher dimensions. Imagine graphing functions not just on a flat 2D plane but in 3D space or beyond. This branch includes various operations such as partial derivatives, multiple integrals, and vector calculus. It is essential for modeling and solving complex problems in physics, engineering, and other fields.

Given a multivariable function, the goal is often to understand how changes in one variable affect the output when others are held constant. This requires careful analysis of each variable's contribution and how they interact. For functions that map to vectors, like in the exercise, each component must be assessed independently. You might deal with planes, curves, or surfaces, each representing a relationship described by the function.

In our exercise, the vector limit is calculated as each component of the vector approaches its specific limit. This demonstrates an important concept in multivariable calculus: often, you need to break down a complex function into simpler parts to understand and solve it.

Vector Limits

Vector limits are a fascinating topic in calculus. They involve understanding what happens to a vector function as the input approaches a certain point. These types of problems often arise in multivariable calculus when analyzing vector fields or curves.

A vector function is a function that outputs a vector rather than a single scalar value. Taking the limit of such a function means analyzing the behavior of each component separately as the input nears a specific point. Then, you recombine these results to find the overall limit of the vector function.

• In the provided exercise, each vector component - exponential, rational, and absolute values - is analyzed individually.
• The limit of a vector is only achieved when the limits of all its components exist and are unique.

The understanding of vector limits allows us to explore the behavior of dynamic systems in fields like physics, where quantities such as velocity and force are naturally expressed as vectors.

Negative Direction Limits

Negative direction limits focus on the behavior of functions as the input approaches a given value from the left or the negative side on the number line. This is important because functions can behave differently when approached from different directions, especially near discontinuities.

In the context of the exercise, the limit as \( t \to 0^- \) requires us to assess the behavior of each function when \( t \) is slightly less than 0. The analysis of limits approaching from the negative side usually involves absolute values and sign reversal, which can be tricky but is crucial in forming a complete understanding of a function’s behavior.

• For example, understanding \( \frac{t}{|t|} \) relies on knowing that \( |t| = -t \) when \( t \lt 0 \).
• Simultaneously, the limit of \( |t| \) translates from negative values to approach zero from the positive side.

Mastering negative direction limits helps in predicting function behavior around critical points and solving real-world problems where directionality matters.