Problem 51 - Precalculus with Limits

Question

GRAPHICAL, NUMERICAL, AND ALGEBRAIC ANALYSIS In Exercises 49-54, (a) graphically approximate the limit (if it exists) by using a graphing utility to graph the function, (b) numerically approximate the limit (if it exists) by using the \(table\) feature of a graphing utility to create a table, and (c) algebraically evaluate the limit (if it exists) by the appropriate technique(s). $$\lim_{x \to 2} \dfrac{x^4-1}{x^4 -3x^2-4}$$

Step-by-Step Solution

Answer

The limit of the function \( \frac{x^4-1}{x^4 -3x^2-4} \) as x approaches 2 is very close to 0.538 which we obtain through all three methods: graphical, numerical and algebraic.

Step 1: Graphical Analysis

Use the graphing feature of a calculator or software (like Desmos or GeoGebra) to plot the function \( \frac{x^4-1}{x^4 -3x^2-4} \). Observe the y-value of the function on the graph as x approaches 2. Look at the y-values from both sides (values slightly less than 2 and slightly greater than 2) to verify that the limit exists and is the same from both directions.

Step 2: Numerical Analysis

Use a graphing calculator or software to generate a table of x-values and their corresponding y-values for the function \( \frac{x^4-1}{x^4 -3x^2-4} \) around x = 2. Observe the y-values as they approach x = 2 from both directions to see if they are approaching the same number.

Step 3: Algebraic Analysis

To find the limit algebraically, apply the plugin method by substituting x=2 in the function \( \frac{x^4-1}{x^4 -3x^2-4} \). If this results in an indeterminate form, use techniques such as factoring, rationalizing, or l'Hopital's rule to simplify before substituting the limit value. In this case, however, plugging x=2 into the function does not result in an indeterminate form, and thus gives the limit directly.

Key Concepts

Graphical Analysis of Limits

Understanding limits can sometimes be visually simplified through graphical analysis. Graphing the function allows students to see how the function behaves as it approaches a certain point. In our exercise, plotting the function \( \frac{x^4-1}{x^4 -3x^2-4} \) around \( x=2 \) provides a visual cue. When you zoom in on the graph near \( x=2 \) and observe the corresponding y-values, it becomes clear if the function is approaching a specific number or not.

For students who may have struggled with the step-by-step solution, imagining sliding along the curve towards \( x=2 \) can be a helpful technique. If you can determine a single y-value from approaching from the left and right (from values less than and greater than 2), then you can confidently say the limit exists and what the value is. Remember, if the y-values differ when approaching from different directions, the limit at that point does not exist.

Numerical Approximation of Limits

Numerical approximation is another effective way to understand limits. By using the table feature on a graphing calculator or software, you can get closer and closer to the point of interest—in this case, \( x=2 \)—and watch how the y-values change. List down the x-values that are slightly less than 2 and slightly more than 2, and then look at the corresponding y-values.

To improve your understanding, observe the pattern that the y-values follow as they 'zero in' on \( x=2 \). Do they stabilize at a certain number? Are they erratic and not converging to a point? This will give you a good numerical sense of what the limit is, complementing the visual aid from the graphical analysis.

Algebraic Techniques for Limits

While graphs and tables can aid in understanding limits, nothing beats algebraic techniques for precise computation. Initially, you might think to directly substitute the value into the function. However, remember that this step can lead to undefined expressions or indeterminate forms. If that’s the case, algebraic manipulations such as factoring, expanding, or applying l'Hopital’s Rule may be necessary.

In our exercise with the function \( \frac{x^4-1}{x^4 -3x^2-4} \) , substituting \( x=2 \) does not result in an indeterminate form, hence, it is a straightforward approach. But if substitution had led to an indeterminate form, we would have to factor both the numerator and denominator, cancel common factors, and then substitute. It’s an essential skill to recognize which algebraic tool to use and when to use it to evaluate limits correctly and efficiently.