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The theory of limits of functions is a significant part of calculus. It is very important for JEE aspirants to have a very clear idea about concepts of limits. Students can expect 2-3 questions from the limits chapter.
Concept of limits
Consider the function f(x) = x3. As x takes values very close to zero, the value of f(x) also moves towards zero. We can say that This is to be read as limit of f(x) as x tends to zero equals zero. In general as x®a, f(x)®l, then l is called limit of function f(x).
Note that there are two ways x could approach a number a either from right or from left. This results in two limits. Right hand limit and left hand limit. The value of f(x) when x tends to a from right is the right hand limit of f(x). It is denoted by . The value of f(x) when x tends to a from left is the left hand limit of f(x). ). It is denoted by . A limit of a function exists only if right hand limit equals left hand limit.
Properties of limits
If f and g be two functions such that both and exists, then
(i)The limit of sum of two functions is the sum of limit of the two functions.i.e., +
(ii)The limit of difference of two functions is the difference of limits of the functions.i.e., –
(iii)The limit of product of two functions is the product of the limits of the functions, i.e., ×
(iv)The limit of quotient of two functions is the quotient of the limits of the functions (whenever denominator is non zero) i.e.,
=
This is the simplest way we can describe limits of functions.
Logarithms
Every positive real number N can be expressed in exponential form as
N = ….(i), where a is a positive real number other than 1. Here a is called the base and x is called the exponent. Example 36 = 62.
We can denote the equation (i) in logarithmic form as . ….(ii) This is read as logarithm of N to the base a equals x.
Here the two equations and are identical where a>0, a≠1 and N>0. Hence logarithm is same as the power to which a number must be raised so as to get some other number . Logarithm of zero does not exist . Logarithm of negative numbers are not defined. Logarithms to the base 10 are written as log , without writing a base down.
Properties of logarithm
If a, b and c are positive real numbers, b≠1 and N is a real number then.
- = +. Multiplication of two logarithmic values is equal to sum of the individual logarithmic values.
- = – . Division of two logarithmic values is equal to the difference of the individual logarithmic values.
- = 0 .The logarithm of 1 to any base is zero.
- = n . The logarithm of a with an exponent is equal to exponent times its logarithm.
- = 1. The logarithm of a number to the same base is 1.
