Textbook Question
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
3. Techniques of Differentiation
Basic Rules of Differentiation
4:32 minutesProblem 84b
Textbook Question
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 guidance1
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}.
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.
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}.
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)}.
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.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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 how functions behave near specific points, which is crucial for defining derivatives and integrals. In this context, evaluating the limit as x approaches 1 allows us to analyze the function's behavior at that point.
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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 question, f'(a) represents the derivative of the function f at the point a, which can be computed using the limit provided.
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L'Hôpital's Rule
L'Hôpital's Rule is a method for evaluating limits that result in indeterminate forms, such as 0/0 or ∞/∞. It states that if the limit of f(x)/g(x) leads to an indeterminate form, the limit can be found by taking the derivative of the numerator and the derivative of the denominator. This rule is particularly useful in the given limit problem, where direct substitution leads to an indeterminate form.
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