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 -
- 1
- 0
- -1
- none of these
The Correct Option is D
Solution and Explanation
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:
limx→0f(x)=undefined.
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Concepts Used:
Continuity
A function is said to be continuous at a point x = a, if
limx→a
f(x) Exists, and
limx→a
f(x) = f(a)
It implies that if the left hand limit (L.H.L), right hand limit (R.H.L) and the value of the function at x=a exists and these parameters are equal to each other, then the function f is said to be continuous at x=a.
If the function is undefined or does not exist, then we say that the function is discontinuous.
Conditions for continuity of a function: For any function to be continuous, it must meet the following conditions:
- The function f(x) specified at x = a, is continuous only if f(a) belongs to real number.
- The limit of the function as x approaches a, exists.
- The limit of the function as x approaches a, must be equal to the function value at x = a.