NIMCET 2014 — Mathematics PYQ

Q: How do I solve NIMCET 2014 Mathematics questions?

Practice NIMCET 2014 Mathematics previous year questions with step-by-step solutions on Mathem Solvex. Each question includes a detailed explanation and video solution to help you master the concept quickly.

Q: What is the difficulty level of NIMCET Mathematics questions?

NIMCET Mathematics questions range from moderate to advanced difficulty. Consistent practice with official previous year papers and topic-wise drills on Mathem Solvex is the most effective preparation strategy.

Q: Are NIMCET previous year papers available for free?

Yes. Mathem Solvex provides free access to NIMCET previous year question papers and topic-wise PDFs. You can download them or attempt them as timed live mock tests directly on the platform.

Question

NIMCET 2014 — Mathematics PYQ

NIMCET | Mathematics | 2014

If $f(x) = \begin{cases} \frac{\sin [x]}{[x]}, & [x] \neq 0 \\ 0, & [x] = 0 \end{cases}$ , where $[x]$ is the largest integer but not larger than $x$ , then $\lim_{x \to 0} f(x)$ is:

Choose the correct answer:

Explanation

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 $[x] = 0$ , $f(x) = 0$ .

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

Substituting $[x] = -1$ :

f(x) = \frac{\sin(-1)}{-1} = \frac{-\sin(1)}{-1} = \sin(1)

\lim_{x \to 0^-} f(x) = \sin(1)

Conclusion:

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

\lim_{x \to 0^-} f(x) \neq \lim_{x \to 0^+} f(x)

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

Correct Option: 4