Problem 7 - Calculus Early Transcendentals

Question

What is the domain of \(f(x)=e^{x} / x\) and where is \(f\) continuous?

Step-by-Step Solution

Answer

Answer: The function \(f(x) = \frac{e^x}{x}\) is continuous on the interval \((-\infty, 0) \cup (0, \infty)\).

Step 1: Identify the Function

The function given is \(f(x) = \frac{e^x}{x}\). This is a rational function where the numerator is an exponential function, \(e^x\), and the denominator is a linear function, \(x\).

Step 2: Find the Domain of the Function

The domain of a function is the set of all possible input values (x-values) for which the function is defined. A rational function is undefined when the denominator is equal to zero. In this case, the denominator is \(x\), so we need to find when \(x\) is equal to zero. So, the function is undefined when \(x=0\). The domain of \(f(x)\) is all real numbers except x = 0. In interval notation, the domain is represented as \((-\infty, 0) \cup (0, \infty)\).

Step 3: Analyze Continuity of the Function

A function is continuous at a point if the value of the function at the point is equal to the limit of the function as x approaches the point. In this case, the exponential function \(e^x\) is continuous for all real numbers, and the linear function \(x\) is also continuous for all real numbers. Since both the numerator and the denominator are continuous for all real numbers, the rational function \(f(x)\) will also be continuous for all real numbers except where the denominator is zero (x=0).

Step 4: State the Function's Continuity

The function \(f(x) = \frac{e^x}{x}\) is continuous on the interval \((-\infty, 0) \cup (0, \infty)\).