UPSC NDA I 2025 Mathematics Paper

Given,
The function isf(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 approaches 0 as x⟶0−..
∴ The value of

lim

x⟶0

f(x)g(x) is 0 .