Problem 33 - 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. $$g(x)=\frac{x^{2}+2 x-3}{x-1} ; x \rightarrow 1$$

Step-by-Step Solution

Answer

The limit is 4 as \(x\) approaches 1.

Step 1: Set up a Table of Values

Create a table with two rows, one for the input values of \(x\) and one for the corresponding output values of \(g(x)\). Choose \(x\) values that get closer to 1 from both sides, such as \(0.9, 0.99, 0.999, 1.001, 1.01, 1.1\).

Step 2: Calculate \(g(x)\) for each \(x\) Value

Substitute each \(x\) value into the function \(g(x) = \frac{x^2 + 2x - 3}{x - 1}\). Compute \(g(0.9), g(0.99), g(0.999), g(1.001), g(1.01), g(1.1)\) and record the results.

Step 3: Observe the Values Approaching a Limit

As \(x\) takes values closer to 1 from both sides, check the corresponding \(g(x)\) values. If these values consistently approach a single number, that number is the limit of the function as \(x\) approaches 1.

Step 4: Write the Limit in Words and Notation

From the table, if the \(g(x)\) values approach a particular number, it indicates that the function approaches a limit as \(x\) approaches 1. Express this by writing, 'The function \(g(x)\) approaches a limit of 4 as \(x\) approaches 1,' and using limit notation as \(\lim_{{x \to 1}} g(x) = 4\).

Key Concepts

Table of Values

A table of values is a helpful tool in understanding functions and their behavior, especially when exploring limits. It involves selecting a series of input values for the variable \(x\) and calculating the corresponding output values of the function.

The process begins by choosing \(x\)-values that incrementally get closer to the point of interest—in this case, \(x = 1\). Pick values both less than and greater than 1, such as \(0.9, 0.99, 0.999, 1.001, 1.01,\) and \(1.1\).

By substituting these \(x\)-values into the function \(g(x) = \frac{x^2 + 2x - 3}{x - 1}\), you can compute and observe the resulting \(g(x)\) values. This table provides a tangible way to visualize how the function behaves as \(x\) approaches 1—essential for grasping the concept of limits.

Function Evaluation

Function evaluation is the act of finding the output of a function for specific input values. In the context of limits, it involves substituting various \(x\)-values into the function to understand its behavior near a specific point.

For the function \(g(x) = \frac{x^2 + 2x - 3}{x - 1}\), evaluate it by substituting values like \(x = 0.9, 0.99, 0.999, 1.001, 1.01,\) and \(1.1\). Simply replace \(x\) in the expression with these numbers and simplify the resulting expressions.

This will give you corresponding \(g(x)\) values that approach the limit. Keep in mind that the closer \(x\) gets to 1, the more accurately these values reflect the function’s behavior near \(x = 1\). This process enables you to hypothesize about the function's limit as \(x\) approaches 1.

Limit Notation

Limit notation is a shorthand used to express the value a function approaches as the input gets close to a particular number. It's an essential part of calculus, providing a formal way to describe the behavior of functions at specific points.

In the exercise, you've computed \(g(x)\) for values near \(x = 1\) and noticed it approaches 4. In limit notation, this is expressed as \(\lim_{{x \to 1}} g(x) = 4\).

This notation has several components:

• The \(\lim\) symbol signifies a limit is being considered.
• The subscript \(x \to 1\) specifies that \(x\) is approaching 1.
• The expression \(g(x)\) is the function being examined.
• Finally, it denotes that the function value is approaching 4 as \(x\) gets closer to 1.

Using this notation effectively communicates the limiting behavior of functions, turning numerical observations into a precise mathematical statement.