1. Limits –
For a function 
the limit of the function at a point 
is the value the function achieves at a point which is very close to .
Formally,
Let be a function defined over some interval containing , except that it
may not be defined at that point.
We say that, 
if there is a number 
for every number 
such that
whenever 
The concept of limit is explained graphically in the following image –
As is clear from the above figure, the limit can be approached from either sides of the number line i.e. the limit can be defined in terms of a number less that 
or in terms of a number greater than . Using this criteria there are two types of limits –
Left Hand Limit – If the limit is defined in terms of a number which is less than then the limit is said to be the left hand limit. It is denoted as 
which is equivalent to 
where 
.
Right Hand Limit – If the limit is defined in terms of a number which is greater than then the limit is said to be the right hand limit. It is denoted as 
which is equivalent to 
where .
Existence of Limit – The limit of a function at exists only when its left hand limit and right hand limit exist and are equal i.e.
Some Common Limits –
L’Hospital Rule –
Example 1 – Evaluate 
· Solution – The limit is of the form 
, Using L’Hospital Rule and differentiating numerator and denominator
· Example 2 – Evaluate 
· Solution – On multiplying and dividing by 
and re-writing the limit we get –
2. Continuity –
· Example 1 – For what value of 
is the function defined by
continuous at 
?
· Solution – For the function to be continuous the left hand limit, right hand limit and the value of the function at that point must be equal.
· 
3. Differentiability –
