Each limit represents a derivative f^'(a). Find f(x) and a. limx →(1)/(4)(x^-1-4)/(x-(1)/(4

step 1 So this problem wants to test your understanding of the definition of a derivative in terms of a

Each limit represents a derivative f^'(a). Find f(x) and a....

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Key Concepts

Derivative Definition via a Limit

This concept explains that the derivative of a function at a point can be defined as a limit. Specifically, it is represented as the limit of the difference quotient as the independent variable approaches a particular value. This definition is foundational in calculus and is used to compute the rate at which the function's value changes at that point.

Difference Quotient

The difference quotient is the expression (f(x) - f(a))/(x - a) where a is a specific point in the domain of the function. This expression represents the average rate of change of the function over an interval, and as the interval shrinks (i.e., x approaches a), it yields the instantaneous rate of change, which is the derivative.

Identifying Functions from Derivative Limits

In problems where a limit is given that is known to represent a derivative at a particular point, the task is to infer both the original function and the point at which the derivative is being taken. This generally involves comparing the given limit with the standard derivative definition, recognizing the structure of the numerator as the difference between f(x) and f(a), and matching it to a known function.

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