Problem 5 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \text { , find the indicated limit. } $$ $$ \lim _{t \rightarrow-1}\left(t^{2}-1\right) $$

Step-by-Step Solution

Answer

The limit is 0.

Step 1: Identify the Function

First, identify the function for which you need to find the limit. Here, the function is given by \( f(t) = t^2 - 1 \).

Step 2: Substitute the Approach Value

Substitute \( t = -1 \) into the function to see how the function behaves as \( t \) approaches \( -1 \).

Step 3: Calculate the Function Value

Substitute \( t = -1 \) into the expression \( t^2 - 1 \). This gives \((-1)^2 - 1\).

Step 4: Simplify the Expression

Calculate \((-1)^2\) which is \(1\), and then subtract \(1\) from it: \(1 - 1 = 0\).

Step 5: Conclude the Limit

Since substituting \(t = -1\) yields \(0\), the limit of \(t^2 - 1\) as \(t\) approaches \(-1\) is \(0\).

Key Concepts

Function Evaluation

In calculus, evaluating a function involves determining the output value of a function when given a specific input. To begin, you need to clearly define the function you're dealing with, which in our exercise, is \( f(t) = t^2 - 1 \). Here, \( t \) is your input variable and you're observing how the function behaves with different values of \( t \). Simply put, function evaluation involves checking what happens to \( f(t) \) as you plug in different values for \( t \). Remember, the first step in tackling such problems is getting familiar with your function's expression, as properly understanding it sets the stage for further calculations and analysis.

Limit Calculation

Calculating limits is a fundamental concept in calculus, which helps understanding how a function behaves as it approaches a certain point. The limit of a function \( f(t) \) as \( t \) approaches a specific value refers to the value that \( f(t) \) gets closer to as \( t \) gets closer to that point.
To calculate limits, follow these basic steps:

• Identify the point that \( t \) is approaching—in this exercise, it's \(-1\).
• Substitute this value into your function, \( f(t) = t^2 - 1 \), to observe the behavior.

The limit of the function \( \lim_{t \to -1} (t^2 - 1) \) is calculated by directly substituting \( t = -1 \). If the function simplifies cleanly, and you do not end up with undefined forms like division by zero, the substituted result is often the limit. In this case, after substitution, the expression simplifies naturally without any further mathematical gymnastics, leading us to our limit value.

Algebraic Simplification

Algebraic simplification is crucial when evaluating limits, as it makes complex expressions more manageable. In our example, substituting \( t = -1 \) into \( t^2 - 1 \) results in \((-1)^2 - 1\).
Simplification here simply involves squaring \(-1\) to get \(1\), followed by subtracting \(1\) to get \(0\).
Here's a quick breakdown:

• First, perform the arithmetic operation: \((-1)^2\) which equals \(1\).
• Then, simplify further: \(1 - 1 = 0\).

This step-by-step simplification helps clear up any confusion and ensures that you arrive at the correct limit value. It highlights the importance of following order of operations, even in seemingly simple expressions, to avoid errors in calculation.