Problem 35 - Precalculus

Question

Use a table of values to evaluate each function as \(x\) approaches the value indicated. If the function seems to approach a limiting value, write the relationship in words and using the limit notation. $$f(x)=3 x^{2}-x-2 ; x \rightarrow 1$$

Step-by-Step Solution

Answer

The limit of \( f(x) \) as \( x \rightarrow 1 \) is 0.

Step 1: Understand the Problem

We need to evaluate the function \( f(x) = 3x^2 - x - 2 \) as \( x \) approaches 1. This involves calculating the value of the function for \( x \) values close to 1 and checking to see if these values approach a particular number.

Step 2: Create a Table of Values

Choose values of \( x \) that are close to 1, including values both less than and greater than 1. For example, \( x = 0.9, 0.99, 0.999, 1.001, 1.01, 1.1 \). Compute \( f(x) \) for these \( x \) values and record the results in a table.

Step 3: Compute Function Values

Compute \( f(x) \) for each chosen \( x \) value.\- \( f(0.9) = 3(0.9)^2 - 0.9 - 2 = 0.43 \)\- \( f(0.99) = 3(0.99)^2 - 0.99 - 2 = 0.0297 \)\- \( f(0.999) = 3(0.999)^2 - 0.999 - 2 = 0.002997 \)\- \( f(1.001) = 3(1.001)^2 - 1.001 - 2 = 0.003003 \)\- \( f(1.01) = 3(1.01)^2 - 1.01 - 2 = 0.0297 \)\- \( f(1.1) = 3(1.1)^2 - 1.1 - 2 = 0.43 \)

Step 4: Analyze the Values

From the table, as \( x \) approaches 1 from both sides, the values of \( f(x) \) are getting closer to 0. This suggests that the function approaches a limiting value of 0 as \( x \) approaches 1.

Step 5: Express the Limit

In words, we can say that as \( x \) gets closer to 1, \( f(x) \) approaches 0. Using limit notation, this is expressed as: \( \lim_{{x \to 1}} f(x) = 0 \).

Key Concepts

Table of Values

When evaluating a function, one way to analyze behavior as a variable approaches a particular value is by using a table of values. This involves selecting various values for the variable that are very close to the point of interest. In this exercise, we're focusing on the function \( f(x) = 3x^2 - x - 2 \) as \( x \) approaches 1.
To create a table of values:

• Choose numbers that are slightly less than 1 and slightly more than 1. In our case, values like 0.9, 0.99, 0.999, 1.001, 1.01, and 1.1 are used.
• Calculate the function's output for each selected \( x \) value.
• Record these results to observe the trend or pattern.

Creating this table helps visually represent the behavior of \( f(x) \) around \( x = 1 \), making it easier to discern any approaching limiting values.

Approaching a Value

In calculus, understanding how a function behaves as it gets close to a particular value of \( x \) is crucial. This involves observing how the function's output changes as the input gets nearer to a specific point. Here, the goal is to evaluate what happens when \( x \) approaches 1 for the function \( f(x) = 3x^2 - x - 2 \).
As \( x \) values are selected closer to 1 from both lower (like 0.999) and higher (like 1.001) sides, the key is to look for a trend in the corresponding \( f(x) \) values. Does \( f(x) \) settle towards a particular number? This pattern suggests the limiting behavior of the function.
In our solution, as \( x \) gets closer to 1, the \( f(x) \) values tend towards 0. This indicates that the function approaches a limiting value of 0 near \( x = 1 \).
This method helps us conclude that there exists a potential "limit" for the function as \( x \) heads toward a specified target.

Limit Notation

Limit notation is a compact and precise way to express the behavior of a function as it approaches a specific point. In calculus, when we say a function "approaches a limit," we are talking about what happens to \( f(x) \) as \( x \) gets infinitely close to some value. For the function \( f(x) = 3x^2 - x - 2 \), as \( x \) nears 1, we found that \( f(x) \) gets closer to 0.
Using limit notation, this relationship can be articulated succinctly as: \[ \lim_{{x \to 1}} f(x) = 0 \]
Breaking down the notation:

• The expression \( \lim \) denotes "limit," indicating that we are looking at the behavior as \( x \) approaches a specific value.
• \( x \to 1 \) shows the point of interest—where \( x \) is headed.
• The function \( f(x) \) is specified, whose behavior we are examining.
• The result, 0, tells us what \( f(x) \) is approaching.

Through this concise framework, limit notation provides a powerful tool for describing function limits clearly and efficiently.