NIMCET 2014 — Mathematics PYQ

Solution:

To find ${\lim }_{x\to 0}f(x)$, we must check the Left Hand Limit (LHL) and the Right Hand Limit (RHL) at $x=0$.

1. Right Hand Limit (RHL):

As $x\to 0^{+}$, $x$ lies in the interval $[0,1)$.

For 0 \leq x < 1, the greatest integer function $[x]=0$.

According to the function definition, when $[x]=0$, $f(x)=0$.

2. Left Hand Limit (LHL):

As $x\to 0^{-}$, $x$ lies in the interval $[-1,0)$.

For -1 \leq x < 0, the greatest integer function $[x]=-1$.

Since $[x]\ne 0$, we use the first part of the function: $f(x)=\frac{\sin [x]}{[x]}$.

Substituting $[x]=-1$:

Conclusion:

Since the Left Hand Limit and Right Hand Limit are not equal:

Therefore, ${\lim }_{x\to 0}f(x)$ does not exist.

Correct Option: 4