Question

If $f\left(x\right) = \frac{\sin\left[x\right]}{\left[x\right]} , \left[x\right]\ne0$ = 0 , [ x ] = 0 Where [x] denotes the greatest integer less than or equal to $x$. then $\displaystyle \lim_{x \to 0} f(x)$ equals -

• A. 1
• B. 0
• C. -1
• D. none of these

Answer 1

Answer: (D)

none of these

Answer 2

The correct option is(D): none of these.
To find the limit of the function f(x)=[x]sin(x)​ as x approaches 0, let's analyze the behavior of the function as x approaches 0 from both the left and the right sides.

As x approaches 0 from the right side (x >0), the greatest integer function [x] is always 0, since all values of x between 0 and 1 are rounded down to 0. Therefore, the function becomes f(x)=0sin(x)​, which is undefined.

As x approaches 0 from the left side (x <0), the greatest integer function [x] is again 0, for the same reason as above. So, the function becomes f(x)=0sin(x)​, which is still undefined.

Since the function is undefined as x approaches 0 from both sides, the limit of f(x) as x approaches 0 does not exist. Therefore, the correct answer is "none of these."

In mathematical terms, this can be formally expressed as:

lim x →0​ f(x)=undefined.