UPSC NDA Math Model Paper 4

1.

lim

x→0

‌sin‌

1

x

⇒−1≤sin‌

1

x

≤1
⇒−1≤

lim

x→0

‌sin‌

1

x

≤1
Here, LHL ≠ RHL, so limit doesn't exist.
2. We have,

lim

x→0

x.sin‌

1

x

Let, x = 1/t, if x → 0, t → 1/x = ∞
∴

lim

t→∞

1

t

.sin‌t=0 (∵ something divide by ∞ = 0)
Here, LHL = RHL, so the limit exists
Hence, option (3) is correct.