UPSC NDA Math Model Paper 1

For a limit to exist Left-hand limit and right-hand limit must be equal.
|x| can have two values
|x | = - x when x is negative
|x| = x when x is positive.

lim

x→0−

|x|

x

=

lim

x→0

−x

x

=−1

lim

x→0+

|x|

x

=

lim

x→0

x

x

=1
Here,

lim

x→0−

|x|

x

≠

lim

x→0+

|x|

x

Hence,

lim

x→0

|x|

x

does not exist.