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🠂1 x¹⁰⁰-1 / x-1

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 f'(a) at a = 1 for the function f(x) = x^{100}.
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) = x^{100} and a = 1, so f(1) = 1^{100} = 1.
3. Step 3: Substitute f(x) = x^{100} and f(1) = 1 into the derivative definition: \lim_{x \to 1} \frac{x^{100} - 1}{x - 1}.
4. Step 4: Notice that the expression \frac{x^{100} - 1}{x - 1} is an indeterminate form 0/0 as x approaches 1. To resolve this, apply L'Hôpital's Rule, which states that if \lim_{x \to c} \frac{f(x)}{g(x)} = \frac{0}{0}, then \lim_{x \to c} \frac{f(x)}{g(x)} = \lim_{x \to c} \frac{f'(x)}{g'(x)}.
5. Step 5: Differentiate the numerator and the denominator separately: f'(x) = 100x^{99} and g'(x) = 1. Then, apply L'Hôpital's Rule: \lim_{x \to 1} \frac{100x^{99}}{1}. Evaluate this limit by substituting x = 1.