Problem 32 - Algebra and Trigonometry Real Mathematics, Real People

Question

Show that the value of \(f(x)\) approaches the value of \(g(x)\) as \(x\) increases without bound (a) graphically and (b) numerically. $$f(x)=1+\left(\frac{3}{x}\right)^{x}, \quad g(x)=e^{3}$$

Step-by-Step Solution

Answer

By applying limit properties and graphical analysis, it can be concluded that the function \(f(x)=1+\left(\frac{3}{x}\right)^{x}\) approaches \(g(x)=e^{3}\) as \(x\) increases without bound.

Step 1: Numerically calculate the limit of \(f(x)\) as \(x\) approaches infinity

Remembering that \(e\) can be defined as the limit \( \lim_{n \to \infty} \left(1+\frac{1}{n}\right)^{n}\), one can see a similarity in the given function \(f(x)\). Thus when taking \( \lim_{x \to \infty} f(x) = \lim_{x \to \infty} 1+\left(\frac{3}{x}\right)^{x} \), the latter part approaches \(e^{3}\) (by substituting \(n = \frac{x}{3}\) into the definition of \(e\)). Therefore, \( \lim_{x \to \infty} f(x) = e^{3}\). Hence numerically, \(f(x)\) approaches \(e^{3}\) as \(x\) increases without bound.

Step 2: Graphical comparison of \(f(x)\) and \(g(x)\)

To confirm graphically, plot both \(f(x)\) and \(g(x)\) on the same graph. As \(x\) increases, the plot of \(f(x)\) will get closer to the horizontal line \(y = e^{3}\), which is the graph of \(g(x)\) considering that it is a constant function. This indicates that \(f(x)\) approaches \(g(x)\) as \(x\) approaches infinity

Step 3: Conclusion

The numerical and graphical evidence support the conclusion that as \(x\) increases, the value of \(f(x)\) approaches the value of \(g(x)\).

Key Concepts

Asymptotic Behavior

Understanding the asymptotic behavior of a function provides insight into how the function behaves as its input grows without bound. In simpler terms, it describes the direction or the value a function approximates as the input variable, often denoted as 'x', becomes very large (tends towards infinity) or very small (tends towards negative infinity).

In the context of our exercise, the function given is \(f(x)=1+\left(\frac{3}{x}\right)^{x}\), and we're asked to show its behavior as x goes to infinity. What this really means is to establish what value \(f(x)\) is getting closer to, as 'x' becomes extremely large. This kind of limit analysis is crucial in understanding long-term trends in a given mathematical model.

Numerical Limit Computation

When dealing with limits, especially when analyzing the behavior at infinity, numerical computation can offer a concrete sense of what's happening. This process involves substituting into the function large values of 'x' to see how the function's value changes.

As we utilize numerical methods for our specific exercise, we remember that the exponential number 'e' can also be understood through a limit process. So, by comparing \(f(x)\) to a known limit representation of 'e', we analyze how the function's output changes as 'x' increases. We observed that as 'x' becomes larger and larger, the expression \((\frac{3}{x})^x\) gets closer to \(e^{3}\), indicating that the limit of the function \(f(x)\) as x approaches infinity is, in fact, \(e^3\).

Graphical Limit Analysis

Graphical analysis is another effective way to visualize the behavior of functions. By plotting \(f(x)\) and \(g(x)\) on the same set of axes, we can directly see how the values of \(f(x)\) approach those of \(g(x)\) as 'x' increases. A graph can tell us, without any computations, whether our function is leveling off or approaching a straight line—a horizontal asymptote—which implies that it’s approaching a specific value.

In this instance, if we plot the given functions, we would see that the curve of \(f(x)\) nears the constant line of \(g(x)\) which is \(y=e^{3}\). The graphical evidence solidly supports our previous numerical findings and gives us a visual confirmation of the limit.

Exponential Functions

Exponential functions are powerful tools in mathematics that grow (or decay) at a rate proportional to their current value. The form of an exponential function is typically seen as \(b^x\), where 'b' is a positive real number. In our exercise, we have \(f(x)\) which involves an exponential component \(\left(\frac{3}{x}\right)^{x}\).

This component displays exponential decay as 'x' grows, since the base \(\frac{3}{x}\) is getting smaller, and hence the entire expression is approaching zero. However, the specific rate at which it does so results in the expression converging to \(e^3\), rather than simply decreasing to zero. Recognizing these behaviors in exponential functions is essential as it helps predict long-term outcomes in various real-world scenarios, such as population growth, radioactive decay, and interest calculations in finance.