Discuss the applicability of Rolle’s theorem for the following functions on the indicated intervals :
f(x) = sin 1/x for – 1 ≤ x ≤ 1
Given function is:
⇒ 
for – 1≤x≤1
Let us check the continuity of the function ‘f’ at the value of x = 0.
We can not directly find the value of limit at x = 0, as the function is not valid at x = 0. So, we take the limit on either sides and x = 0, and we check whether they are equal or not.
Right – Hand Limit:
⇒ 
We assume that the limit 
, kϵ[ – 1,1].
⇒ 
, where h>0
⇒ 
⇒ 
⇒ 
...... (1)
Left – Hand Limit:
⇒ 
⇒ 
, where h>0
⇒ 
⇒ 
⇒ 
⇒ 
⇒ 
......(2)
From (1) and (2), we can see that the Right hand and left – hand limits are not equal, so the function ‘f’ is not continuous at x = 0.
∴ Rolle’s theorem is not applicable to the function ‘f’ in the interval [ – 1,1].