Question:
Let [ ] denote the greatest integer function and f (x) = [tan$^2$ x]. Then
Let [ ] denote the greatest integer function and f (x) = [tan$^2$ x]. Then
Updated On: Sep 3, 2024
lim f (x) (for x \(\to\) 0) does not exist
- f (x) is continuous at x = 0
- f (x) is not differentiable at x = 0
- f (x) = 1
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The Correct Option is B
Solution and Explanation
Check the continuity of the function
f (x) = [tan$^2$ x] at x = 0.
L.H.L. (at x = 0)
= $\underset{\text{x $\rightarrow$ 0$^-$}}{{lim }}$[tan$^2$ x] = $\underset{\text{h $\rightarrow$ 0}}{{lim }}$[tan$^2$(0 - h)]
= $\underset{\text{h $\rightarrow$ 0}}{{lim }}$[tan$^2$ h] = [tan$^2$ 0] = [0] = 0
R.H.L. (at x = 0)
= $\underset{\text{x $\rightarrow$ 0$^+$}}{{lim }}$[tan$^2$ x] = $\underset{\text{h $\rightarrow$ 0}}{{lim }}$[tan$^2$(0 - h)]
= $\underset{\text{h $\rightarrow$ 0}}{{lim }}$[tan$^2$ h] = [tan$^2$ 0] = [0] = 0
Now, determine the value of f(x) at x = 0.
f (0) = [tan$^2$ 0] = [0] = 0
Hence, f (x) is continuous at x = 0.
f (x) = [tan$^2$ x] at x = 0.
L.H.L. (at x = 0)
= $\underset{\text{x $\rightarrow$ 0$^-$}}{{lim }}$[tan$^2$ x] = $\underset{\text{h $\rightarrow$ 0}}{{lim }}$[tan$^2$(0 - h)]
= $\underset{\text{h $\rightarrow$ 0}}{{lim }}$[tan$^2$ h] = [tan$^2$ 0] = [0] = 0
R.H.L. (at x = 0)
= $\underset{\text{x $\rightarrow$ 0$^+$}}{{lim }}$[tan$^2$ x] = $\underset{\text{h $\rightarrow$ 0}}{{lim }}$[tan$^2$(0 - h)]
= $\underset{\text{h $\rightarrow$ 0}}{{lim }}$[tan$^2$ h] = [tan$^2$ 0] = [0] = 0
Now, determine the value of f(x) at x = 0.
f (0) = [tan$^2$ 0] = [0] = 0
Hence, f (x) is continuous at x = 0.
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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.