Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \lim _{t \rightarrow 1}\left[\frac{t-1}{t^{2}-1} \mathbf{i}-\frac{t^{2}+2 t-3}{t-1} \mathbf{j}\right] $$

Step-by-Step Solution

Answer

The limit is \( \frac{1}{2} \mathbf{i} - 4 \mathbf{j} \).

Step 1: Simplify the First Component

The first component of the vector limit expression is \( \frac{t-1}{t^2-1} \mathbf{i} \). Notice that \( t^2 - 1 \) can be factored as \( (t-1)(t+1) \). Therefore, the expression becomes: \[ \frac{t-1}{(t-1)(t+1)} \mathbf{i} = \frac{1}{t+1} \mathbf{i}, \text{ for } t eq 1. \]

Step 2: Simplify the Second Component

The second component of the vector is \( -\frac{t^2 + 2t - 3}{t-1} \mathbf{j} \). Factor \( t^2 + 2t - 3 \) as \((t+3)(t-1)\). Thus it becomes: \[ -\frac{(t+3)(t-1)}{t-1} \mathbf{j} = -(t+3) \mathbf{j}, \text{ for } t eq 1. \]

Step 3: Rewrite the Vector

After simplifying both components, the vector expression is: \[ \frac{1}{t+1} \mathbf{i} - (t+3) \mathbf{j}. \]

Step 4: Evaluate the Limit

Substitute \( t = 1 \) into the simplified vector component expressions. For the \( \mathbf{i} \) component: \( \frac{1}{1+1} = \frac{1}{2} \). For the \( \mathbf{j} \) component: \( -(1+3) = -4 \).

Step 5: Combine to Find the Limit

The limit of the vector as \( t \to 1 \) is: \[ \frac{1}{2} \mathbf{i} - 4 \mathbf{j}. \]

Key Concepts

Limits of Vector Functions

Understanding limits in the context of vector functions is crucial for calculus students. Vector functions can be thought of as functions that yield a vector as output rather than a scalar. In this particular exercise, we were required to find the limit of a vector function as the variable approaches a specified point. Here's a quick rundown on why and how we use limits for vectors:

• We evaluate the limit of each component of the vector separately, just as you would with a scalar function.
• If each of the vector's components has a limit as the variable approaches a point, then the vector function overall also has a limit.
• Note that the function must be simplified before substituting the limit point, especially if substituting directly gives an indeterminate form like \( \frac{0}{0} \).

In our case, we simplified each component of the vector before substituting \( t = 1 \) to ensure an accurate limit was calculated.

Factoring Expressions

Factoring expressions is a common technique used in calculus, particularly in evaluating limits. It involves rewriting expressions as a product of simpler expressions or factors, making them easier to manipulate. This comes in handy especially when you're facing indeterminate forms.How did factoring help in this exercise?

• For the \( \frac{t-1}{t^2-1} \) component, the denominator \( t^2-1 \) was factored as \( (t-1)(t+1) \).
• This allowed the \( t-1 \) terms to cancel, simplifying the expression and avoiding division by zero, which leads to an indeterminate form.
• Similarly, factoring \( t^2 + 2t - 3 \) as \( (t+3)(t-1) \) simplified the second component by cancelling \( t-1 \).

Thus, factoring not only streamlined the function but also facilitated the evaluation of the limit.

Rational Functions

Rational functions are ratios of polynomials. They hold significance in calculus for their diverse properties, particularly when analyzing limits. In this exercise, we dealt with rational expressions, which demanded careful simplification before evaluating their limits. Key takeaways on rational functions:

• Because they are ratios, they often involve factoring to simplify the expression and eliminate possible points of indeterminacy.
• Asymptotic behavior, where the function approaches a default value as the variable nears infinity or a specific point, is a crucial aspect to consider.
• They frequently require additional techniques like factoring or multiplying by the conjugate for limits evaluation to simplify the path to a solution.

In the given exercise, understanding the structure of the rational expressions enabled the breaking down of complex limits into simpler, manageable forms for each component of the vector.