The Limit Definition Of Derivatives And Its Importance In Calculus

Limit Definition of Derivative

limit (as h approaches 0)= F(x+h)-F(x)/h

The limit definition of the derivative of a function f(x) at a point x=a is given by:

f'(a) = lim_{h->0} (f(a+h) – f(a))/h

Intuitively, the derivative f'(a) measures the rate at which the function f(x) changes at the point x=a. To compute the derivative using the limit definition, we take a small change in x, denoted by h, and compute the corresponding change in the function f(x), denoted by f(a+h) – f(a). We then divide this change by the size of the change in x, which is h, to get the average rate of change of the function over the interval [a, a+h]. Finally, we take the limit as h approaches 0 to obtain the instantaneous rate of change of the function at x=a.

Another way to interpret the limit definition of the derivative is that it gives the slope of the tangent line to the graph of the function f(x) at the point x=a. As h approaches 0, the two points on the graph of f(x) that determine the slope of the tangent line become arbitrarily close together, so the limit of the difference quotient gives the slope of the tangent line at x=a.

It is important to note that not all functions have derivatives at every point. In particular, a function may not have a derivative at a point if the limit in the definition does not exist or is infinite. In addition, some functions may have derivatives at some points but not at others, and the derivative may also be differentiable from one side but not from the other at a given point.

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