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Limit of a function at a point.

published on February 16th, 2008 . by Vanaja

The notion of limit is one of the most basic and powerful concepts in all of mathematics. Differentiation and Integration, which comprise the core of study in calculus, are both products of the limit. The concept of limit is the foundation stone of calculus and as such is the basis of all that follows it. It is extremely important that you get a good understanding of the notion of limit of a function if you have a desire to fully understand calculus at the entry level.

It is very important that you get a good understanding of the notion of limit of a function if you have a desire to fully understand calculus.

Definition

Let f(x) be a function of x. Let a and l be constants such that as x\rightarrow a, we have f\left(x \right)\rightarrow l. In such case we say that the limit of the function f(x) as x approaches a is l. we write this as

\lim_{x\rightarrow a}f\left(x \right)=l

In case , no such number l exist, then we say that \lim_{x\rightarrow a}f\left(x \right)does not exist finitely.

Illustration

Let a regular polygon of n sides be inscribed in a circle. The area of the polygon cannot be greater than the area of the circle., however large the number of sides of the polygon increases indefinitely the area of the polygon continually approaches the area of the circle. Thus the difference between the area of the circle and the polygon can be made as small as we please by sufficiently increasing the number of sides of the polygon.

We have \lim_{n\rightarrow \infty } (Area of the polygon of n sides)=Area of the circle.

Limits

published on January 9th, 2007 . by Vanaja

The notion of limit is one of the most basic and powerful concepts in all of mathematics. Differentiation and Integration, which comprise the core of study in calculus, are both products of the limit. The concept of limit is the foundation stone of calculus and as such is the basis of all that follows it. It is extremely important that you get a good understanding of the notion of limit of a function if you have a desire to fully understand calculus at the entry level.

You can find some basic results on limits here

Periodic Functions

published on November 16th, 2006 . by Vanaja

Periodic functions are functions that repeat its values over and over, after some definite period or cycle on a specific period. This can be expressed mathematically that A function f is said to be periodic if there exists a real T>0 such that f (x+T) = f(x) for all x.

The fundamental period of a function is the length of a smallest continuous portion of the domain over which the function completes a cycle. That is, it’s the smallest length of domain that if you took the function over that length and made an infinite number of copies of it, and laid them end to end, you would have the original function.

If a function is periodic, then the smallest t>0 ,if it exists such that f (x+t) = f(x) for all x, is called the fundamental period of the function.

The trigonometric functions sine and cosine are common periodic functions, with period 2?.
ie. sin (x+2?)= sin x , cos(x+2?)=cos x

But tan and cot remain unchanged when x is increased by pi.
ie. tan(x+?)=tan x, cot(x+?)= cot x
So, they are periodic functions with period ? .

An aperiodic function (non-periodic function) is one that has no such period

Range and Domain of Functions

published on October 4th, 2006 . by Vanaja

Yesterday we have learned what is a function.
Today let’s discuss about range and domain of a function.


Answers


Try to do more problems from your text.

Functions

published on October 3rd, 2006 . by Vanaja

Today we can discuss a topic from functions.

Definitions:

Function
Let A anb B be two non empty sets. A function “f” from a set A to a set B is a rule so that to each element x in A there corresponds exactly one element y in B, under f ,then we say that f is a functin from A to B and write
f:A -> B

y is called the image of x under f and is denoted by f(x). x and y are respectively called the independent variable and the dependent variable. We also say that y is a function of x and write
y=f(x)

Examples:

  1. In the family of circles, the area A of the circle is a function of radius r of the circle.
    Here radius r is the independent variable and area A is the dependent variable
  2. The speed of a chemical reaction increases 2 times with the addition of every 5 milligrams of a catalyst. Here the amount of catalyst is the independent variable and speed of the chemical reaction is the dependent variable.