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).

Key Concepts

Limits

A limit is a fundamental concept in calculus that describes the behavior of a function as its input approaches a certain value. It helps in understanding the function's behavior near points of interest, including points of discontinuity or where the function is not explicitly defined. Evaluating limits is essential for determining derivatives and integrals.

Derivatives

The derivative of a function at a point measures 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 over an interval as the interval approaches zero. In this context, f'(a) represents the instantaneous rate of change of the function f at the point a.

Exponential Functions

Exponential functions, such as e^x, are functions where the variable appears in the exponent. The function e^x is particularly important in calculus due to its unique property that its derivative is equal to itself. Understanding the behavior of e^x as x approaches 0 is crucial for evaluating the given limit and finding f'(a).