If the functions f(x) and g(x) are continuous at x=a then,

1> is continuous at x=a.

2> is continuous at x=a.

3> is continuous at x=a , if g(x) is not equal to 0.

4> is continuous at x=a if f(x) is greater than 0 , or is a positive number when “n” is even.

Such curve or line can be drawn by the continuous motion of a pencil in a sheet of paper.

And Discontinuous  function is just opposite of the continuous function , the function “f” is said to be  discontinuous function when the graph drawn for f(x) is  consists of disconnected curves or lines.

For example:

Continuous Function:

Discontinuous Function:

If we zoom into the disconnected place of two curves it looks like:

When x_a is any point in the  interval [a,b] then the following relation should be true in the curve for the function “f” to be a continuous function if the following relation is not true in the curve then it is a discontinuous function.

1> The limit of the sum (or difference) of the functions “f” and “g” is the sum (or difference) of the limits of the functions

i.e.

2> The limit of the product of the function “f” and “g” is the product of the limits of the functions.

i.e.

3> The limit of the quotient of the function “f” and “g” is the quotient of the limits of the functions.

i.e.

4> The limit of n^th root of a function “f” is the n^th root of the limit of the function.

i.e.

and x∈ (a-β , a+β)

And let a function f(x) be defined at the Interval (a-β , a+β) .

Then we can also find the limit of  function f(x) as,

Left hand Limit of a Function:

In the above  case the limit of f(x) when “x” approaches “a” from the left hand side of the interval is known as the left hand limit of f(x).

and is denoted by:

Right hand Limit of a Function:

Similarly,

the limit of f(x) when “x” approaches “a” from the right hand side of the interval is known as the right hand limit of f(x) and is denoted by:

A set of points lying between any two points “a” and “b” is known as an Interval.

For example:

the interval from points x=1 to x=10 is the set of points lying between 1 and 10.

Open and Closed Interval:

An Interval which includes it’s end points(a and b)  is known as Closed interval ; While an Interval Which does not includes it’s end points(a and b)  is known as Open Interval.

A open Interval from point “a” to point “b” is denoted by:

(a,b)

and  A closed Interval from point “a” to point “b” is denoted by:

[a,b]

For example:

The open and closed interval from point “a” to “b” where a=1 and b=5 are:

(a,b)={2,3,4}

[a,b]={1,2,3,4,5}

First study the following picture:

You can see in the figure ; if a polygon is inscribed inside a circle , then

as we increase the number of sides of polygon the area of polygon gradually increases .

As we increase the number of sides in the polygon it’s area gradually approaches

the area of  circle and if we take a polygon with sufficiently large number of sides in same configuration

then , the area of the polygon will become almost equal to the area of circle  , but

it will never be equal to   the area of the circle.

In this case , the limit or limiting value of the area of polygon when the number of sides in polygon tends to infinity  is the area of circle which encloses it.

mathematically we can denote this case as:

If the number of sides in the polygon is denoted by “n”  , the function which defined the area of polygon is  f(n) and the area of circle which encloses the polygon is denoted by “A” then :

Similarly we can denote this case of limit in any function an example is given below:

If a function is defined by f(x)=x+2

Then if we limit the value of  ”x” to “1″ then the value of the function “f” approaches 1+2=3

Which can be shown as: