Question

Numerical and Graphical Analysis In Exercises 3-8, use a graphing utility to complete the table and estimate the limit as \(x\) approaches infinity. Then use a graphing utility to graph the function and estimate the limit graphically. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & 10^{0} & 10^{1} & 10^{2} & 10^{3} & 10^{4} & 10^{5} & 10^{6} \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ $$ f(x)=\frac{-6 x}{\sqrt{4 x^{2}+5}} $$

Short answer

It might be found that as \(x\) approaches infinity, the function \(f(x)=\frac{-6 x}{\sqrt{4 x^{2}+5}}\) seems to approach a limit of -1.5, which can be concluded both from the table and the graph.

Step-by-step solution

Complete the table

Using a graphing utility, evaluate the function \(f(x)=\frac{-6 x}{\sqrt{4 x^{2}+5}}\) at each specified point (from \(10^{0}\) to \(10^{6}\)) to get respective \(f(x)\) values. Fill the respective output into the table.

Estimate the limit numerically

Based on the table obtained in step 1, estimate the limit of \(f(x)\) as \(x\) approaches infinity. This can be done by observing the trend of the function values as \(x\) increases. Typically, if the values of \(f(x)\) seem to be approaching a particular number, that number would be the limit.

Graph the function and estimate the limit graphically

Use the graphing utility to graph the function \(f(x)=\frac{-6 x}{\sqrt{4 x^{2}+5}}\). The value that the function seems to be approaching as \(x\) goes to infinity on the graph also represents the limit.

Key concepts

Limits at Infinity

When exploring the behavior of a function as the input values grow larger and larger, we are dealing with limits at infinity. Essentially, this concept helps us understand what happens to a function's output when the input, or in mathematical terms, the variable, is infinitely large. An important point is that limits at infinity do not always exist, but when they do, they can indicate the function's eventual behavior or its horizontal asymptote.

For example, when examining the limit of the function \( f(x) = \frac{-6x}{\sqrt{4x^2+5}} \) as \( x \) approaches infinity, you're predicting where the function's values will trend towards as \( x \) gets very large. If the values of \( f(x) \) settle near a certain number, this number would be considered the function's limit at infinity. Effectively, this concept is central to understanding the long-term behavior of functions in calculus.

Numerical Analysis of Limits

The numerical analysis of limits involves assessing the limit of a function by looking at its values at specific points, particularly as those points get very large (or very small in the case of approaching negative infinity). In the step-by-step solution for estimating the limit, you begin by filling out a table for \( f(x) = \frac{-6x}{\sqrt{4x^2+5}} \) at various values of \( x \).

By inputting these increasingly larger values of \( x \), you create a sequence of function values that provide insight into the function's behavior. If these values start to converge or stabilize around a constant value, this gives us a numerical estimation of the limit. For instance, if as \( x \) increases, the value of \( f(x) \) begins to level off around -3, we would say that the limit of \( f(x) \) as \( x \) approaches infinity is -3. This numerical approach is a straightforward yet powerful method for estimating a function's limit.

Graphical Analysis of Limits

A graphical analysis of a function offers a visual representation of its behavior and is an invaluable tool for estimating limits. After plotting the function \( f(x) = \frac{-6x}{\sqrt{4x^2+5}} \) using a graphing utility, you would observe the curve's trajectory as \( x \) increases. The trend you're looking for is where the function's values are heading.

If the graph of the function flattens out and approaches a horizontal line as \( x \) gets larger, the y-coordinate of this line will give you a graphical estimation of the limit. This method is particularly useful because it provides a visual context for the limit, aiding in comprehension and reinforcing the concept learned through numerical analysis. By combining both numerical and graphical analyses, you can gain a robust understanding of limits.

Use of Graphing Utilities in Calculus

Graphing utilities, such as graphing calculators or software programs, are integral tools in the field of calculus. They empower students and mathematicians to visualize complex functions and easily estimate their limits, both at infinity and at specific points. In the context of our function \( f(x) = \frac{-6x}{\sqrt{4x^2+5}} \), a graphing utility allows you to quickly create precise graphs and tables.

By effortlessly plotting the function or generating values for a sequence of inputs, a graphing utility simplifies the process of analysis, which might otherwise be very time-consuming or challenging to perform by hand. Through the use of these technologies, calculus becomes more accessible and comprehensible, enabling students to explore and confirm their results in multiple ways.