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

Key Concepts

Domain of a Function

The domain of a function refers to the set of all possible input values (x-values) for which the function is defined. For the function f(x) = e^x/x, we need to identify values of x that do not lead to undefined expressions, such as division by zero. In this case, the function is undefined at x = 0, so the domain is all real numbers except zero.

Continuity of a Function

A function is continuous at a point if the limit of the function as it approaches that point equals the function's value at that point. For f(x) = e^x/x, we must check continuity at all points in its domain. Since the function is undefined at x = 0, it is not continuous there, but it is continuous for all other real numbers.

Exponential Functions

Exponential functions, such as e^x, are characterized by a constant base raised to a variable exponent. They are defined for all real numbers and exhibit rapid growth. In the context of f(x) = e^x/x, the exponential component contributes to the function's behavior as x approaches positive or negative infinity, influencing its overall continuity and limits.