Question

Use a graphing utility to graph the function and estimate the limit (if it exists). What is the domain of the function? Can you detect a possible error in determining the domain of a function solely by analyzing the graph generated by a graphing utility? Write a short paragraph about the importance of examining a function analytically as well as graphically. $$ \begin{array}{l} f(x)=\frac{e^{x / 2}-1}{x} \\ \lim _{x \rightarrow 0} f(x) \end{array} $$

Short answer

The limit of the function as \( x \rightarrow 0 \) is approximately \( 0.5 \), and the domain of the function is all real numbers except \( x=0 \). The graphing method might not clearly show the function being undefined at \( x=0 \), reinforcing the importance of both analytical and graphical examination of functions.

Step-by-step solution

Graphing The Function

First, graph the function \( f(x)=\frac{e^{x/2}-1}{x} \) using a graphing utility. Note how the function behaves as \( x \) approaches 0.

Estimating The Limit

Estimate the limit of the function as \( x \) approaches 0 by looking at the behavior of the function on the graph near the point \( x=0 \). The limit of the function as \( x \rightarrow 0 \) appears to be \( 0.5 \) based on the graph.

Determining The Domain

The domain of a function is the set of all values of \( x \) for which the function is defined. \( f(x) \) is undefined for \( x=0 \) because there's a division by \( x \). Therefore, the domain of the function is all real numbers except \( x=0 \). From the graph, the function is defined for all values of \( x \) except \( x=0 \). This is a potential error in determining the domain of a function by analyzing the graph alone, since the graph might not clearly show the function is undefined at \( x=0 \).

Importance Of Analytic And Graphic Examination

Examining a function both analytically and graphically is important because while graphs provide visual information about the function's behavior, they can sometimes miss certain properties, like points of discontinuity. Analytical examination, on the other hand, provides precise information and can reveal properties that might not be noticeable on the graph. Therefore, it's always best to combine both forms of examination to get a complete understanding of a function.

Key concepts

Limit of a Function

Understanding the limit of a function is a fundamental concept in calculus. It describes the behavior of a function as it approaches a particular point, but not necessarily reaching it. For example, the exercise presents us with the task of estimating the limit of the function \( f(x) = \frac{e^{x/2}-1}{x} \) as \( x \) approaches 0. Graphically, we can observe that as \( x \) gets infinitely close to 0, \( f(x) \) seems to approach 0.5.

However, graphical estimation may not always be precise. That's why an analytical approach, typically involving algebraic manipulation or special limit laws, serves as a crucial tool to accurately determine limits. In particular, L'Hôpital's rule often helps find limits that involve indeterminate forms like \( \frac{0}{0} \). When \( x \) is exactly 0, \( f(x) \) is undefined, which is why the analytical examination is a must to complement the graphical one and confirm that the limit exists and equals 0.5.

Function Domain

The domain of a function is the set of all possible input values (usually represented as \( x \) values) for which the function is defined. Determining the domain correctly is essential for understanding the scope within which the function operates. In our exercise example, \( f(x) = \frac{e^{x/2}-1}{x} \) is not defined when \( x = 0 \) because division by zero is undefined in mathematics. Thus, the domain of \( f(x) \) includes all real numbers except 0.

When using a graphing utility, the domain might appear continuous if the gap at the excluded point is too small to notice. This is a pitfall that reinforces the value of analyzing a function algebraically to accurately establish its domain.

Analytical Examination of Functions

An analytical examination of functions delves into the nuts and bolts of a function using algebraic techniques. This form of analysis does more than just identify the limit or domain; it also uncovers the function's continuity, differentiability, and behavior at asymptotes. The precision that comes from this analysis allows mathematicians and students alike to predict and describe a function’s behavior with high accuracy.

For example, understanding the behavior of \( f(x) \) near \( x = 0 \) would involve looking at the numerator and denominator separately and determining how they interact as \( x \) approaches 0. By doing so, one can confirm findings from graphing utilities and avoid errors that arise from a purely visual interpretation of a function.

Graphing Utility in Calculus

A graphing utility is an invaluable tool in calculus that visually presents the behavior of functions. By graphing \( f(x) \) across a range of values, one can quickly approximate characteristics such as limits, asymptotes, and intercepts. With the exercise function \( f(x) = \frac{e^{x/2}-1}{x} \) graphed, students can intuitively see the trends and estimate the limit as \( x \) approaches 0.

However, one should note the graphing utility's limitations, such as precision errors or the inability to show certain discontinuities. These shortcomings must be covered by an analytical approach. This is why graphing is often used as a first step in understanding a function’s behavior before conducting a more detailed analytical examination.