## Limit Definition of a Derivative at a Point

### The limit definition of a derivative at a point is a mathematical formula that is used to calculate the instantaneous rate of change of a function at a specific point

The limit definition of a derivative at a point is a mathematical formula that is used to calculate the instantaneous rate of change of a function at a specific point. The derivative of a function represents the slope of the tangent line to the graph of the function at a given point.

The limit definition of a derivative at a point can be defined mathematically as follows:

Let f(x) be a function and let ‘a’ be a specific point at which we want to find the derivative. The derivative of f(x) at point ‘a’ is given by:

f'(a) = lim(x->a) [f(x) – f(a)] / (x – a)

This formula can be read as the limit of the difference quotient as x approaches a. The difference quotient calculates the average rate of change between two points on a function, and by taking the limit as the points get infinitesimally close, we find the instantaneous rate of change at the specific point ‘a’.

To use this formula, we substitute the function values ‘f(x)’ and ‘f(a)’ into the formula, and then evaluate the limit as x approaches ‘a’. The result of this limit calculation will give us the value of the derivative at point ‘a’, representing the slope of the tangent line to the function at that point.

It is important to note that the limit exists only if the function is continuous at ‘a’ and the function is differentiable at ‘a’. Differentiability implies that the function does not have any sharp corners or vertical tangents at the given point. If the limit exists, it provides a precise calculation of the instantaneous rate of change at that point.

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