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.