MHTCET 2016 Math Solved Paper

(a) Given, f(x) = {

x‌sin‌ 1 x	for‌x≠0
k	for‌x=0

As f(x) is continuous at x=0
So, LHL =RHL=f(0) ......(i)
Now, LHL = limx→0−f(x) = lim‌f(0−h)
=limh→0(0−h)‌sin‌

1

(0−h)

= limh→0(−h)‌sin(−

1

h

)
= limh→0h‌sin‌

1

h

=0 × finite value lies between − 1 and 1=0
[∵limh→0‌sin‌

1

h

= finite value lies between −1‌and‌1]
and f(0)=k
Now, from eq. (i), LHL = f ( 0)
⇒ 0 = k
Hence, k= 0