Functions Limits and Continuity


   
 
Limits
 
 
But we are interested in finding the value of y near 2.
 
 
When
 
x = 1.9, y = 3.9
 
x = 1.99, y = 3.99
 
x = 1.999 y = 3.999
 
. .
 
. .
 
. .
 
Similarly, when
 
x = 2.1 y = 4.1
 
x = 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
 
 
 
 
 
 
Limit of composite functions
 
 
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
 
 
If P(c) = a, Q(c ) = 0, then
 
 
Example:
 
 
At x =1, the numerator of the rational function 6, but denominator is 0.
 
 
 
Theorem 1:
 
 
Proof:
 
Case (i): If n is a positive integer.
 
We have,
 
 
 
 
 
 
Case (ii): If n is a negative integer.
 
Let n = -k, where k is a positive integer
 
 
 
 
 
 
Hence true for negative integers.
 
Case (iii): When n is a rational number.
 
 
 
 
 
 
 
 
 
 
 
 
Hence the theorem.
 
 
     
   
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