Limit Definition of a Derivative at a Point
f'(a) = lim x→a ([f(x)-f(a)]/[x-a])
The limit definition of a derivative at a point is a mathematical concept used to determine the instantaneous rate of change of a function at a specific point. The concept of a derivative and its limit definition are fundamental to calculus.
The definition of the derivative at a point x=a is given by:
f’(a) = lim_(h→0) [f(a + h) – f(a)]/h
Where f’(a) is the derivative of the function f at the point a. The limit is taken as h approaches 0, which means that we are looking at the behavior of the function f near the point a.
The numerator of the fraction, f(a + h) – f(a), represents the change in the value of the function over a small interval of length h, centered around the point a. The denominator, h, represents the length of this interval.
Thus, the fraction ([f(a + h) – f(a)]/h) represents the average rate of change of the function over the interval [a, a + h]. The limit of this fraction as h approaches 0 gives the instantaneous rate of change of the function at the point a, which is the derivative.
The limit definition of a derivative at a point can be used to calculate the derivative of any function that is differentiable at that point. By approaching the limit as h approaches 0, we can get a very accurate estimate of the derivative at that point, and thus a very accurate understanding of the behavior of the function near that point.
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