LIMITS :
Let y = f(x) be a given function defined in the neighbourhood of x = a, but not necessarily at the point x = a.
Basically we are interested in examining the behaviour (or tendency) of the function f(x) when the distance between the points x and a is small, i.e. is small but not zero.
The limiting behavior of the function in the neighbourhood of x = a when |x–a| is small is called the limit of the function when x approaches a.
Mathematically we write this as limx->0 f(x).
Let limx->0 f(x) = l . It would simply mean that when we approach the point x = a from the values which are just greater than or just smaller than x= a, f(x) would have a tendency to move closer to the value ‘ l ’ .
This is same as saying, “difference between f(x) and l can be made as small as we feel like by suitably choosing x in the neighbourhood of x = a”.
Mathematically, we write this as, limx->0 f(x) = l , which is equivalent to saying that
such that 0 < |x-a|< δ and ε depends on δ .
Where ε and δ are sufficiently small positive numbers.
It is clear from the above discussion that if we are interested in finding the limit of f(x) at x = a, the first thing that we have to make sure is that f(x) is well defined in the neighbourhood of x = a and not necessarily at x = a
(that means x = a may or may not be in the domain of f(x)), because we have to examine it’s behavior or tendency in the neighbourhood of x = a.
Now the following possibilities may arise;
(a) Left tendency of f(x) is same as it’s right tendency, as shown in the adjacent figure.
That means when we approach x= a from the values which are just less than a , f(x) has a tendency to move towards the value ‘ l ’ (left tendency).
Similarly when we approach x = a from the values which are just greater then a, f(x) has a tendency to move towards the value ‘ l ’ (right tendency).
In this case we say f(x) has limit at x = a i.e. limx->0 f(x) = l
(b) When the left tendency is not the same as the right tendency, as shown in the adjacent figure.
Clearly left tendency (l1) is not the same as right tendency ( l2). In this case we say limit of f(x) at x = a will not exist.
(c) When the left tendency and/or the right tendency is not fixed, (as shown in the adjacent figure).
It is clear that in this case, the function has erratic behavior in the neighbourhood of x = a, and it will not be possible to talk about the left and the right tendencies of the function in the neighbourhood of x = a.
In this case we conclude that the limit of f(x) at x = a will not exist.
From the above mentioned discussion, it is clear that the limit of f(x) at x = a would exist if and only if f(x) is well defined in the neighbourhood of x = a (not necessarily at x = a) and has a unique behaviour in the neighbourhood of x = a.
If any of these conditions are not fulfilled the limit of f(x) at x = a will not exist.
Left and Right Limit
Let y = f(x) be a given function, and x = a is the point under consideration. Left tendency of f(x) at x = a is called it’s left limit and right tendency is called it’s right limit.
Left tendency (left limit) is denoted by f(a − 0) or f(a −) and right tendency (right limit) is denoted by f( a + 0) or f(a +) and are written as
where ‘ h ‘ is a small positive number.
Thus for the existence of the limit of f(x) at x = a, it is necessary and sufficient that
f(a − 0) = f(a + 0), if these are finite or f(a − 0) and f(a + 0) both should be either + ∞ or − ∞
Frequently used Series Expansions :
Following are some of the frequently used series expansions:
Illustration 1.
Evaluate the following limits, if exist
Solution :
Thus the given limit does not exist.
