Problem 3 - College Algebra and Calculus: An Applied Approach

Question

Complete the table and use the result to estimate the limit. Use a graphing utility to graph the function to confirm your result. $$ \lim _{x \rightarrow 2} \frac{x-2}{x^{2}-4} $$ $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 1.9 & 1.99 & 1.999 & 2 & 2.001 & 2.01 & 2.1 \\ \hline f(x) & & & & ? & & & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The limit of the function as x approaches 2 is 0.5.

Step 1: Simplify the Function

Simplify by factoring the denominator, \(x^{2}-4\). This is a difference of squares and can be factored into \((x-2)(x+2)\), so the function simplifies to \( \frac{x-2}{(x-2)(x+2)} \). Since x is not equal to 2, you can cancel out \(x-2\) from the numerator and denominator, which results in \( \frac{1}{x+2} \).

Step 2: Find the Value of the Function

Plug in the given x-values (1.9, 1.99, 1.999, 2.001, 2.01, 2.1) into the simplified function \( \frac{1}{x+2} \). This will give you the corresponding f(x) values for the table.

Step 3: Estimate the Limit

Observe the tendency of f(x) values as x gets closer to 2 from both sides (left and right). The limit is the value that the function approaches as x nears 2.

Step 4: Graph the Function and Confirm the Result

Using a graphing utility, graph the function \( \frac{1}{x+2} \). The y-value that the function approaches at x=2 will confirm the limit previously estimated.

Key Concepts

Difference of squares

The expression you have here is \( x^2 - 4 \), which is known as a difference of squares. This is a special type of polynomial where two perfect squares are separated by a subtraction operator. To recognize a difference of squares, look for expressions that fit the pattern \( a^2 - b^2 \), where both \( a \) and \( b \) are integers and are perfect squares. By factoring a difference of squares, you can break it down into \((a-b)(a+b)\). In your given function, \( x^2 - 4 \) simplifies to \((x-2)(x+2)\), as 4 is the same as \(2^2\). This factoring is crucial for simplifying expressions and solving limits.

Factoring

Factoring is an essential step in simplifying complex expressions, especially when working with limits. In the problem you've tackled, factoring turns the denominator \( x^2 - 4 \) into the product \((x-2)(x+2)\). This enables us to see components of the function that can be potentially canceled. \[(x^2 - 4 = (x-2)(x+2))\] Why is this important? If \( x \) is not equal to 2, you can cancel out the \( x-2 \) from the numerator and denominator. This simplification reduces the function to \( \frac{1}{x+2} \), which can be more easily evaluated as \( x \) approaches 2. Essentially, factoring simplifies the expression to focus on the behavior of the function as it nears the point of interest.

Graphical verification

Once you've simplified the function, verifying your result graphically is a powerful tool. Graphical verification involves using a graphing utility to confirm the behavior of the function visually. For the function \( \frac{1}{x+2} \), plot the graph and observe its path as \( x \) heads towards 2. The graph should clearly illustrate the y-values as \( x \) approaches 2, affirming the limit you've calculated in earlier steps. The graph provides a visual confirmation of the function's limit, depicting how the function approaches its limit without needing complex calculations. This visual approach helps you understand the concept of limits by seeing the behavior in action.

Function simplification

Simplifying a function is key when dealing with limits, especially when the expression is initially undefined at a point. In the exercise, simplifying \( \frac{x-2}{x^2-4} \) into \( \frac{1}{x+2} \) is pivotal. This simplification makes it possible to compute the limit as \( x \) approaches 2. Importantly, the simplification removes the undefined form - making it manageable. Without simplification, substituting \( x = 2 \) would yield an undefined result. By removing the \( x-2 \) term from both the numerator and the denominator, you eliminate the discontinuity and can then examine the true limiting behavior of the function as \( x \) nears the point of discontinuity. This step shows how simplification can convert a complex limit problem into one that’s easy to understand.