Problem 68 - Thomas Calculus

Question

You will find a graphing calculator useful. Let $$g(x)=\left(x^{2}-2\right) /(x-\sqrt{2}).$$ a. Make a table of the values of \(g\) at the points \(x=1.4,1.41\) , \(1.414,\) and so on through successive decimal approximations of \(\sqrt{2} .\) Estimate \(\lim _{x \rightarrow \sqrt{2}} \sqrt{2} g(x).\) b. Support your conclusion in part (a) by graphing \(g\) near \(x_{0}=\sqrt{2}\) and using Zoom and Trace to estimate \(y\) -values on the graph as \(x \rightarrow \sqrt{2}\) . c. Find \(\lim _{x \rightarrow \sqrt{2}} g(x)\) algcbraically.

Step-by-Step Solution

Answer

The limit is \(2\sqrt{2}\).

Step 1: Understanding the Function

The function given is \( g(x) = \frac{x^2 - 2}{x - \sqrt{2}} \). We need to find the limit of this function as \( x \) approaches \( \sqrt{2} \).

Step 2: Creating a Table of Values

Calculate the values of \( g(x) \) at \( x = 1.4, 1.41, 1.414, ... \) to estimate \( \lim_{x \to \sqrt{2}} g(x) \). Substitute these values into the equation. For example: \( g(1.4) = \frac{1.4^2 - 2}{1.4 - \sqrt{2}} \). Continue this for increasingly close approximations to \( \sqrt{2} \).

Step 3: Analyzing the Table of Values

Examine the pattern in the values from the table. As \( x \) gets closer to \( \sqrt{2} \), observe whether \( g(x) \) appears to be approaching a specific value.

Step 4: Graphical Analysis

Graph the function \( g(x) \) using a graphing calculator. Use the Zoom and Trace features to see the behavior of the graph as \( x \) approaches \( \sqrt{2} \). Note the \( y \)-value the graph approaches as \( x \) nears \( \sqrt{2} \).

Step 5: Algebraic Simplification

Simplify \( g(x) \) algebraically. Notice that \( x^2 - 2 = (x - \sqrt{2})(x + \sqrt{2}) \), so the expression can be written as \( g(x) = x + \sqrt{2} \) for \( x eq \sqrt{2} \).

Step 6: Finding the Limit Algebraically

Using the simplified expression from Step 5, as \( x \to \sqrt{2} \), \( g(x) = x + \sqrt{2} \), thus \( \lim_{x \to \sqrt{2}} g(x) = \sqrt{2} + \sqrt{2} = 2\sqrt{2} \).

Key Concepts

Graphical Analysis

Graphical analysis provides visual insights into the behavior of a mathematical function. By plotting the function \(g(x) = \frac{x^2 - 2}{x - \sqrt{2}}\) on a graph, we can observe how \(g(x)\) behaves as \(x\) approaches \(\sqrt{2}\). Visualizing the function can help confirm our calculations.
Using a graphing calculator, utilize the "Zoom" feature to closely inspect the function's behavior around \(x = \sqrt{2}\). As you trace along the graph, pay attention to how the \(y\)-value changes when \(x\) gets nearer to \(\sqrt{2}\). This tool allows you to see the tangent-like behavior around that point, which is crucial when estimating the limit.
The graphical representation not only offers a different perspective but also aids in detecting possible discontinuities or unique function characteristics near the limit point. Its visual aspect complements the numeric and algebraic approaches, confirming that the function should approach a specific \(y\)-value as \(x\) nears \(\sqrt{2}\).

Table of Values

Creating a table of values for \(g(x)\) as \(x\) gets closer to \(\sqrt{2}\) helps to numerically estimate the limit. This step involves computing \(g(x)\) for values such as \(x = 1.4, 1.41, 1.414,\) and so forth.
Each entry in the table represents a finer approximation of \(\sqrt{2}\), allowing you to observe a pattern in how \(g(x)\) behaves. Here's an approach to fill out the table:

• Calculate \(g(1.4) = \frac{1.4^2 - 2}{1.4 - \sqrt{2}}\).
• Compute similar values for increasingly precise approximations of \(\sqrt{2}\).
• Identify the trend seen in these calculated values.

With the numbers from the table, you get to see how \(g(x)\) approaches or hovers around a particular value as \(x\) approaches \(\sqrt{2}\).
In this specific exercise, you would expect the table values to suggest \(g(x)\) is approaching \(2\sqrt{2}\) or something similar. This hint of stability or convergence gives confidence in your approximation before any algebraic simplification is done.

Algebraic Simplification

Algebraic simplification plays a vital role in determining the limit value analytically. The function \(g(x) = \frac{x^2 - 2}{x - \sqrt{2}}\) can be simplified by recognizing factoring opportunities.
First, apply the identity \(x^2 - 2 = (x - \sqrt{2})(x + \sqrt{2})\), allowing us to write the function as:\[g(x) = \frac{(x - \sqrt{2})(x + \sqrt{2})}{x - \sqrt{2}}\]For \(x eq \sqrt{2}\), cancel the \(x - \sqrt{2}\) terms, simplifying \(g(x)\) to:\[g(x) = x + \sqrt{2}\]
Now, finding \(\lim_{x \to \sqrt{2}} g(x)\) involves substituting \(x = \sqrt{2}\) into this simplified function, resulting in:\[\lim_{x \to \sqrt{2}} g(x) = \sqrt{2} + \sqrt{2} = 2\sqrt{2}\]
This algebraic approach eliminates the indeterminate form seen at the original function's limit, providing clear confirmation of the limit's value. It also demonstrates the power of algebraic techniques to transform complex expressions into simpler, equivalent forms that reveal underlying truths about the function's behavior as \(x\) approaches \(\sqrt{2}\).