The following limits represent f'(a) for some function f and some real number a.
Find a possible function f and number a.
lim x🠂0 e^x-1 / x

Verified step by step guidance

1. Step 1: Recognize that the given limit \( \lim_{x \to 0} \frac{e^x - 1}{x} \) is a standard limit that represents the derivative of the exponential function \( f(x) = e^x \) at a specific point.
2. Step 2: Recall the definition of the derivative \( f'(a) \) for a function \( f(x) \) at a point \( a \), which is given by \( \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \).
3. Step 3: Compare the given limit with the derivative definition. Notice that the form \( \frac{e^x - 1}{x} \) suggests \( f(x) = e^x \) and \( f(a) = e^0 = 1 \).
4. Step 4: Identify that the limit \( \lim_{x \to 0} \frac{e^x - 1}{x} \) is equivalent to finding \( f'(0) \) for the function \( f(x) = e^x \).
5. Step 5: Conclude that a possible function \( f \) is \( f(x) = e^x \) and the number \( a \) is 0, since the derivative \( f'(0) \) is represented by the given limit.

Key Concepts

Limits

Limits are fundamental in calculus, representing the value that a function approaches as the input approaches a certain point. In this context, the limit as x approaches 0 of (e^x - 1) / x is crucial for determining the derivative of the function at that point. Understanding limits allows us to analyze the behavior of functions near specific values, which is essential for defining derivatives.

Derivatives

The derivative of a function at a point quantifies the rate at which the function's value changes as its input changes. It is defined as the limit of the average rate of change of the function as the interval approaches zero. In this case, the limit provided represents the derivative of the function f at the point a, which is a key concept in understanding how functions behave locally.

Exponential Functions

Exponential functions, such as e^x, are functions of the form f(x) = a^x, where 'a' is a constant. The function e^x is particularly important in calculus due to its unique property that its derivative is equal to itself. This property simplifies the process of finding derivatives and limits involving exponential functions, making them a common subject in calculus problems.