Problem 52 - 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). $$\lim_{x \to 2} \dfrac{x^4-2x^2-8}{x^4 -6x^2+8}$$

Step-by-Step Solution

Answer

In graphical analysis, observe the graph at point x = 2. For numerical analysis, create a table of values near x = 2. For algebraic analysis, substitute x = 2 in the limit expression to find that it equals \(-0.571\).

Step 1 - Graphical analysis

Using any graphing utility, enter the given function \(\dfrac{x^4-2x^2-8}{x^4 -6x^2+8}\) and observe its value at x = 2. This will give a graphical approximation of the limit.

Step 2 - Numerical analysis

Using the table function of a graphing utility, create a table showing values of the function at points close to x = 2. This will give a numerical approximation of the limit.

Step 3 - Algebraic analysis

Rearrange the limit expression to easier form if possible. If not, simply substitute the value of x = 2 into the limit function to get the exact value. The limit is then \(\lim_{x \to 2} \dfrac{x^4-2x^2-8}{x^4 -6x^2+8} = \dfrac{(2)^4-2(2)^2-8}{(2)^4 -6(2)^2+8}\)

Key Concepts

Graphical Analysis

Graphical analysis is a powerful visual way to understand limits. To find a limit graphically, you use a graphing tool to plot the function. This gives you a picture of the function's behavior as it approaches a certain value. In our exercise, the function is \(\dfrac{x^4-2x^2-8}{x^4 -6x^2+8}\). By graphing this function, you can see what happens as \(x\) approaches 2. This visual approach helps confirm whether the function has a limit at that point or if it perhaps approaches infinity or has a gap. By observing the graph near \(x = 2\), you will get an approximate value of the limit—look at whether the curve flattens out towards a specific number as it comes closer to \(x = 2\).

Numerical Approximation

Numerical approximation involves creating a table of values for the function around the point of interest, here it is \(x = 2\). This is done using the table feature of a graphing calculator or software. By inputting values incrementally closer to 2, you can see how the function behaves. If the values in your table get closer to a specific number as \(x\) approaches 2, that number is the numerical approximation of the limit. For instance, choosing values like 1.9, 1.99, 1.999, 2.001, 2.01, and 2.1 will provide insight into the trend of the function as \(x\) gets closer to 2. Analyzing the trend of these values helps confirm your graphical observation and provides a numerical estimate of what the limit might be.

Algebraic Evaluation

Algebraic evaluation is about calculating the exact value of the limit using mathematical techniques. This step involves simplifying the expression or substituting the \(x\) value directly into the function if simplification is not possible. In some cases, rearranging the expression can help find a common factor or apply limit laws that make calculation easier. For our function, substitute \(x = 2\) directly to evaluate: \[\lim_{x \to 2} \dfrac{x^4-2x^2-8}{x^4 -6x^2+8} = \dfrac{2^4-2(2)^2-8}{2^4 -6(2)^2+8}\]Simplifying this results in \(\dfrac{0}{0}\), which is an indeterminate form requiring further work. Factorization or L'Hôpital's Rule could be among the techniques used to resolve such indeterminate forms, aiming at a more manageable form through which a limit can be explicitly found.