For Exercises 55 and 56, evaluate each limit by first converting each to a derivative at a particular x-value.

lim (x → 1) (x⁵⁰ − 1) / (x − 1)

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Recognize that the given limit \( \lim_{x \to 1} \frac{x^{50} - 1}{x - 1} \) resembles the definition of a derivative. Specifically, it can be seen as the derivative of a function at a point.

Recall the definition of the derivative: \( f'(a) = \lim_{x \to a} \frac{f(x) - f(a)}{x - a} \). Here, \( f(x) = x^{50} \) and \( a = 1 \).

Identify that the expression \( x^{50} - 1 \) can be rewritten as \( f(x) - f(1) \), where \( f(x) = x^{50} \) and \( f(1) = 1^{50} = 1 \).

Apply the derivative definition: The limit \( \lim_{x \to 1} \frac{x^{50} - 1}{x - 1} \) is equivalent to finding \( f'(1) \), where \( f(x) = x^{50} \).

Differentiate \( f(x) = x^{50} \) to find \( f'(x) = 50x^{49} \). Then, evaluate \( f'(1) = 50 \times 1^{49} \) to find the value of the limit.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Limit

A limit in calculus is the value that a function approaches as the input approaches a certain point. In this problem, we are interested in the behavior of the function (x⁵⁰ − 1) / (x − 1) as x approaches 1. Understanding limits is crucial for analyzing the behavior of functions at points where they might not be explicitly defined.

Derivative

The derivative of a function at a point provides the rate at which the function's value changes as its input changes. It is the slope of the tangent line to the function at that point. In this problem, converting the limit into a derivative involves recognizing that the expression resembles the definition of a derivative, specifically the derivative of x⁵⁰ at x = 1.

L'Hôpital's Rule

L'Hôpital's Rule is a method for evaluating limits of indeterminate forms, such as 0/0. It states that if the limit of f(x)/g(x) as x approaches a point results in an indeterminate form, the limit can be found by taking the derivative of the numerator and the denominator separately. This rule is applicable here to simplify the limit expression by differentiating the numerator and denominator.

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