Problem 85 - Precalculus with Limits

Question

Use a graphing utility to graph \(f\) and \(g\) in the same viewing window and determine which is increasing at the greater rate as \(x\) approaches \(+\infty .\) What can you conclude about the rate of growth of the natural logarithmic function? $$\begin{array}{ll}{\text { (a) } f(x)=\ln x,} & {g(x)=\sqrt{x}} \\ {\text { (b) } f(x)=\ln x,} & {g(x)=\sqrt[4]{x}}\end{array}$$

Step-by-Step Solution

Answer

Without the exact graphs, a precise answer can't be given. However, in general, the growth rate of the natural logarithmic function is slower than both the square root and fourth root function as \(x\) approaches \(+\infty\). Therefore, \(g(x) = \sqrt{x}\) and \(g(x) = \sqrt[4]{x}\) would increase at a greater rate than \(f(x) = \ln x\) as \(x\) approaches \(+\infty\).

Step 1: Graph the functions

In the first instance, graph the functions \(f(x)=\ln x\) and \(g(x)=\sqrt{x}\) in the same viewing window. Note the behaviour of both graphs as \(x\) approaches \(+\infty\). In the second instance, do the same process but now with the functions \(f(x)=\ln x\) and \(g(x)=\sqrt[4]{x}\).

Step 2: Compare the rates of growth

Once you have the graphs, observe which function is increasing at a greater rate as \(x\) approaches \(+\infty\). For both instances, note whether it is \(f(x)\) or \(g(x)\) that is growing more rapidly.

Step 3: Analyze the rate of growth of the logarithmic function

Based on the comparison from Step 2, make a conclusion regarding the rate of growth of the natural logarithmic function. Specifically, you should be able to identify whether the logarithmic function grows faster or slower than the square root function, and how this behaviour changes for the fourth root function. This will contribute to the overall understanding of how the natural logarithmic function behaves.

Key Concepts

Graphing Functions

When we tackle mathematical problems involving functions, visualizing them becomes a crucial step. Graphing functions, which means plotting them on a coordinate plane, gives us a picture of the equation's behaviour across different values.

In the exercise, graphing the functions allows us to compare the rates at which the natural logarithmic function, represented as \(f(x)=\text{ln } x\), and the square root and fourth root functions, represented as \(g(x)=\sqrt{x}\) and \(g(x)=\sqrt[4]{x}\) respectively, increase. By examining the graphs, one can discern how steep the functions are at different points, especially as \(x\) tends toward positive infinity. This visual approach is instrumental in understanding the underlying relations and trends of these functions.

Natural Logarithmic Function

The natural logarithmic function, notated as \(\text{ln } x\), is an increasing function with a special base, \(e\), which is an irrational constant approximately equal to 2.71828. This function is the inverse of the exponential function \(e^x\), and it plays a fundamental role in various math and science fields.

Its graph shows distinctive features such as passing through the point \((1,0)\), since the logarithm of 1 is 0 for any base, and exhibiting an asymptotic behaviour as \(x\) approaches zero—it never touches the x-axis. Its rate of growth is initially fast but slows down as \(x\) increases, which contrasts with algebraic functions like square roots, where the growth rate is the opposite.

Asymptotic Behavior

Asymptotic behavior refers to the behavior of curves as they approach a particular line or point but never actually reach it. In graphing, an asymptote is this line.

For example, the natural logarithmic function \(\text{ln } x\) has an asymptote at \(x=0\), where the function approaches negative infinity but never crosses the y-axis. This concept is vital for understanding limits and can explain how functions behave toward the extremes of their domains. By learning how to identify asymptotes and the behavior of functions near them, students can gain a deeper understanding of the function's long-term behavior, which is often unveiled when graphing it.

Increasing Functions

An increasing function is one where the output, or y-value, increases as the input, or x-value, increases. They can be steadily increasing, such as linear functions with a positive slope, or increase at varying rates, such as the natural logarithmic function.

When comparing \(f(x)=\text{ln } x\) with \(g(x)=\sqrt{x}\) and \(g(x)=\sqrt[4]{x}\), we observe the rate at which each function is increasing. The exercise reveals that while all three functions are increasing, they do so at different rates as \(x\) approaches infinity—a crucial concept to understand the dynamics of each function's growth. Intuitively, algebraic functions like roots grow without bound, but logarithmic growth tends to slow, providing an interesting contrast in the behaviors of different types of increasing functions.