x = 1.9, y = 3.9
x = 1.99, y = 3.99x = 1.999 y = 3.999
. .. .
. .Similarly, when
x = 2.1 y = 4.1x = 2.01 y = 4.01
x = 2.001 y = 4.001. .
. .. .
Observe that as x approaches 2, y approaches 4. 4 is called the limit of f(x) as x approaches 2 and is represented as
- If x is close to 2, f(x) is close to 4.
- If |x - 2| is small then |f(x) - 4| is also small.
Note 2:
When x approaches 2 from left, the limit obtained is called left hand limit and when x approaches 2 from the right, the limit obtained is called the right hand limit.
Note 3:
We say that 
because both left hand limit and
right hand limit are finite and equal.
If there exists a real number l such that if |f (x) - l| can be made as small as we possible by taking x sufficiently close to a, then l is called the limit of f (x) as x tends to 'a'.
Left Hand Limit
Let f(x) tend to a limit l1 as x tends to a through values less than 'a', then l1 is called the left hand limit which is symbolically written as
Right Hand Limit
Let f(x) tend to a limit l2 as x tends to 'a' through values greater than 'a', then l2 is called the right hand limit which is symbolically written as
Note:
We say that limit of f(x) exists at x = a, if l1 and l2 are both finite and equal.
Some Properties of Limits
Limits of Polynomial Functions and Rational Functions
a)
Limits of polynomial functions can be found by substitution
If f(x) = anxn + an-1xn-1+an-2xn-2+…….. a0, then
Example:
b)
Limit of a rational function can be found by substitution, if the limit of the denominator is not zero.
In other words,
where P(x) and Q(x) are polynomials, then we have
Example:
c)
If Q(c)=0, common factors between the numerator and denominator are identified and cancelled. This reduces the rational function to another rational function whose denominator is not zero at c.
Example:
If Q(c) = 0, we can also factorise and cancel a common factor.
Example:
Suggested answer:
d)
In the rational function, we have
Example:
Theorem 1:
Proof:
Case (i): If n is a positive integer.
We have,
Let n = -k, where k is a positive integer
Hence true for negative integers.
Case (iii): When n is a rational number.
