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

검증된 단계별 안내

Step 1: Identify the domain of the function.

Step 2: Recognize that the function f(x) = \(\frac{e^x}{x}\) is a rational function.

Step 3: Determine where the denominator is zero, since division by zero is undefined.

Step 4: Set the denominator x equal to zero and solve for x.

Step 5: Conclude that the domain of f(x) is all real numbers except x = 0, and f is continuous on its domain.

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.