Question

The limit of \(f(x)=(1+x)^{1 / x}\) is a natural base for many business applications, as you will see in Section \(10.2\). \(\lim _{x \rightarrow 0}(1+x)^{1 / x}=e \approx 2.718\) (a) Show the reasonableness of this limit by completing the table. $$ \begin{array}{|l|l|l|l|l|l|l|l|} \hline x & -0.01 & -0.001 & -0.0001 & 0 & 0.0001 & 0.001 & 0.01 \\ \hline f(x) & & & & & & & \\ \hline \end{array} $$ (b) Use a graphing utility to graph \(f\) and to confirm the answer in part (a). (c) Find the domain and range of the function.

Short answer

The table completion and graph of the function \(f(x)=(1+x)^{1 / x}\) confirms that the limit as x approaches 0 is approximately 'e' or 2.718. The domain of the function is all real numbers excluding zero (-inf^, +inf^), and the range is (0, inf^).

Step-by-step solution

Completing the table

To complete the table, substitute each given value of \(x\) into the function \(f(x)=(1+x)^{1 / x}\) to find the corresponding output: \nFor instance, when \(x = -0.01\), \(f(x) = (1 - 0.01)^{1 / (-0.01)} = 2.7048\) (approx). Likewise, repeat this for other values of \(x\) to fill the table.

Approximating the limit

By observing the computed \(f(x)\) values in the table, it can be seen that as x goes close to zero, the value of \(f(x)\) approaches 'e' \( \approx 2.718\). This confirms that the limit of the function as \(x\) approaches 0 is 'e', as stated in the problem.

Graphing the function

Graph the function \(f(x)=(1+x)^{1 / x}\) using a graphing tool or calculator. It should display that as \(x\) goes close to zero, \(f(x)\) approaches a y-value of 'e', or approximately 2.718. This further confirms the limit stated in the exercise.

Determine the domain and range of the function

The domain of a function is the set of all possible \(x\) values which will make the function 'work', and will output real \(y\) values. Here \(x\) can be any real number excluding zero (-inf^, +inf^) because at \(x=0\), the function is undefined. \nThe range of a function is the set of all possible output values (y-values), which come out from the function after we have substituted the domain. Since the function never reaches zero and its limiting value approaching zero from either side is 'e', the range is (0, inf^).

Key concepts

Natural Base e

In calculus, the natural base, denoted as \(e\), is an irrational number approximately equal to 2.718. It’s named the "natural" base because of the way it arises naturally in various growth and decay processes, such as in compound interest calculations and population dynamics.
Understanding \(e\) is fundamental because it serves as the base for natural logarithms, which are logarithms to the base \(e\). These logarithms have significant applications in fields ranging from science to finance. For example:

• In exponential growth models, such as population growth, \(e\) helps describe how quickly quantities grow based on a continuous rate.
• In finance, \(e\) is used in the calculation of compound interest over continuous periods, maximizing returns on investments.

The limit expression \(\lim_{x \to 0}(1+x)^{1/x} = e\) shows how \(e\) arises from a simple exponential structure. This natural occurrence makes \(e\) a pivot around which much of calculus revolves.

Domain and Range

The concepts of domain and range are essential when analyzing any function. Let's break these down for this exercise's function, \(f(x) = (1 + x)^{1/x}\).

• The domain of a function refers to all possible \(x\) values that the function can accept without causing any undefined behaviors, such as division by zero or square roots of negative numbers. Here, \(x\) cannot be zero because \((1 + x)^{1/x}\) is undefined due to the division by zero. Thus, the domain is all real numbers excluding zero or \(( -\infty, 0) \cup (0, +\infty )\).
• The range of a function is the set of possible output values. For this function, as \(x\) approaches 0 from both directions, the function approaches \(e\), but never reaches any value that would be zero or negative. Thus, all positive real numbers are possible outputs, so the range is \((0, +\infty)\).

Understanding domain and range helps identify the limits of a function’s behavior and is crucial for correctly interpreting its graph and real-world applications.

Calculating Limits

Limits in calculus are the foundation for defining derivatives and integrals. They describe how a function behaves as it approaches a particular point. A limit calculation involves the precise determination of what value a function approaches as the input gets closer to some value.
For the function \(f(x)=(1+x)^{1/x}\), calculating the limit as \(x\) approaches zero is a classic problem that helps define the number \(e\). Since directly substituting \(x=0\) in this function is not possible (due to the zero in the denominator), we approach with values close to zero. For instance, calculating at \(x = -0.001\) and \(x = 0.001\) to approximate that value ever closer to \(e\).
Limits can be calculated using:

• Substitution, when the point does not cause undefined behavior.
• Simplification techniques to manage indeterminate forms.
• L'Hôpital's Rule, which deals with 0/0 forms in limits by differentiating the numerator and denominator.

Effective calculation and understanding of limits are essential as they allow us to find tangents to curves and calculate areas under curves, laying the groundwork for more advanced calculus concepts.