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Calculus is that branch of mathematics which mainly deals with the study of change in the value of a function as the points in the domain change.

Limits

A limit is the value that a function approaches as the input approaches some value.

Let f(x) be a function define on a domain containing some interval, whenever a x → a we say limit of f(x) is l.

Left Hand and Right Hand Limit

There are essentially two ways x could approach a number a either from left or from right. All the values of x near a could be less than a or could be greater than a. This leads to two limits - the right hand limit and the left hand limit.

The expected value of the function as dictated by the points to the left of a point defines the left hand limit of the function at that point. Similarly the right hand limit. Limit of a function at a point is the common value of the left and right hand limits, if they coincide.

Algebra of Limits

Suppose limx→a f(x) = α and limx→a g(x) = β then we can define the following rules:

 

Standard Limits

Continuity

A function from the set of real numbers can be represented by a graph in the cartesian plane. The function is continuous if the graph is a single unbroken curve.

Differentiability

Any differentiable function must be continuous at every point in its domain. The converse does not hold. A continuous function need not to be differentiable.

Derivative of a function f at any point xis defined by

Standard Derivatives