UPSC NDA I 2025 Mathematics Paper

Given,
The function is f(x)=sin‌(⌊x⌋), where ⌊x⌋ is the greatest integer function, and g(x)=|x|, the absolute value function.
We are tasked with finding:

lim

x⟶0

‌

f(x)

g(x)

For g(x)=|x|, we know that:

lim

x⟶0

g(x)=0
For f(x)=sin‌(⌊x⌋), we know that:
For x⟶0+,⌊x⌋=0, so f(x)=sin‌(0)=0.
For x⟶0−,⌊x⌋=−1, so f(x)=sin‌(−1), which is a nonzero constant.
Evaluating the limit:
For x⟶0+,‌

f(x)

g(x)

=‌

0

x

=0
For x⟶0−,‌

f(x)

g(x)

=‌

sin‌(−1)

−x

, which becomes undefined as x⟶0−because the denominator approaches 0 , but the numerator remains a nonzero constant.
∴ Since the left-hand and right-hand limits do not match, the limit does not exist.