The following limits represent f'(a) for some function f and some real number a.
b. Evaluate the limit by computing f'(a).
lim x🠂0 e^x-1 / x

Verified step by step guidance

1. Step 1: Recognize that the given limit represents the derivative of a function at a point. Specifically, it is the definition of the derivative of the function f(x) = e^x at the point a = 0.
2. Step 2: Recall the definition of the derivative: f'(a) = lim_{x \to a} \frac{f(x) - f(a)}{x - a}. In this case, f(x) = e^x and a = 0, so f(0) = e^0 = 1.
3. Step 3: Substitute the function and the point into the derivative definition: f'(0) = lim_{x \to 0} \frac{e^x - 1}{x}.
4. Step 4: Recognize that this is a standard limit that can be evaluated using L'Hôpital's Rule, which applies when the limit is in the indeterminate form 0/0. L'Hôpital's Rule states that lim_{x \to c} \frac{f(x)}{g(x)} = lim_{x \to c} \frac{f'(x)}{g'(x)} if the limit is indeterminate.
5. Step 5: Differentiate the numerator and the denominator: the derivative of e^x is e^x, and the derivative of x is 1. Apply L'Hôpital's Rule: lim_{x \to 0} \frac{e^x}{1}. Evaluate this limit to find f'(0).