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.
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:
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.
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