Limits and Continuity


Suppose f (x) is a real valued function m is a real number. The expression

$$\lim_{x \rightarrow c} f(x)= l$$

This means that f( x ) can be taken very close to l. For example let us consider the function f( x ) in the

$$f(x) = \frac{x^2-1}{x-1}$$

So if you consider x=0.9 then f( 0.9 ) = $\frac{0.9^2 -1}{0.9-1}$ = 1.900. So, if you try some more values like

1. f ( 0.99 ) = 1.990,
2. f ( 0.999 ) = 1.999
3. f ( 1 ) = undefined.
4. f( 1.001 ) = 2.001
5. f( 1.01 ) = 2.010.

Limit Concept

In the above situation, if we take the limit from lower values of x and the other from higher values of x then the function reaches the same value which is approximately equal to 2. so here we can say that the left hand limit is equal to right hand limit and also we say that the limit exists.

1. so f(x) = l = RHS lim exists x- c
2. f(x) = l =LHS lim exits,

Therefore, it is clear that the limit will exists only when right hand side limit is equal to left hand side limit.



Function y = f(x) is continuous at point x=b if the following three conditions are satisfied:

1. f(b) is defined ,

2. lim f( x ) exists

Lim f( x ) exists = f(b) is defined ,

Function f is said to be continuous on an interval I if f is continuous at each point x in I. Here is a list of some well-known facts related to continuity:

1. The sum is continuous
2. The difference is continuous
3. The PRODUCT of 2 continuous functions are continuous
4. The quotient of continuous functions is continuous at all points, given that denominator is not zero.
5. The composition of continuous functions is continuous at all points.
6. Any polynomial is continuous for all values of x.
7. Function ex, sinx, cosx are all continuous for all values of x.

The graph below shows the function which is continuous.

Continuous Function

The graph of a discontinuous function

Discontinuous Function

Next Chapters

Number Theory Linear Equation Set Theory Math Fractions
Math Functions Pyramid Calculus Cone
Cylinder Chain Rule Limits and Continuity Prime Factorization
Square Roots and Cube Roots Parabola Distance Formula Definite Integrals
Interest Simple Interest Compound Interest Area of Irregular Figures

Latest Articles

Average Acceleration Calculator

Average acceleration is the object's change in speed for a specific given time period. ...

Free Fall Calculator

When an object falls into the ground due to planet's own gravitational force is known a...


In Mathematics, the permutation can be explained as the arrangement of objects in a particular order. It is an ordered...

Perimeter of Rectangle

A rectangle can be explained as a 4-sided quadrilateral which contains equal opposite sides. In a rectangle

Perimeter of Triangle

A three sided polygon which has three vertices and three angles is called a triangle. Equilateral triangle...