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The value of lim x->0 [sinx/x], where [.] represents the greatest integer function is a) 1 b) 0 c) -1 d) none of the above The correct answer is option 'b', can you please explain this answer?
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The value of lim x->0 [sinx/x], where [.] represents the greatest inte...
Solution:
Given, lim x->0 [sinx/x], where [.] represents the greatest integer function.
Let's evaluate the limit by considering the left-hand limit and right-hand limit separately.
Left-hand limit:
As x approaches 0 from the negative side, sinx/x takes negative values.
As sinx oscillates between -1 and 1, [sinx/x] takes values from -2 to 0.
Therefore, the left-hand limit is -1.
Right-hand limit:
As x approaches 0 from the positive side, sinx/x takes positive values.
As sinx oscillates between -1 and 1, [sinx/x] takes values from 0 to 2.
Therefore, the right-hand limit is 0.
Since the left-hand limit and right-hand limit are not equal, the limit does not exist.
Hence, the correct option is none of the above.
Note: The greatest integer function cannot be applied to a limit. The limit should be evaluated first and then the greatest integer function can be applied to the result.
Given, lim x->0 [sinx/x], where [.] represents the greatest integer function.
Let's evaluate the limit by considering the left-hand limit and right-hand limit separately.
Left-hand limit:
As x approaches 0 from the negative side, sinx/x takes negative values.
As sinx oscillates between -1 and 1, [sinx/x] takes values from -2 to 0.
Therefore, the left-hand limit is -1.
Right-hand limit:
As x approaches 0 from the positive side, sinx/x takes positive values.
As sinx oscillates between -1 and 1, [sinx/x] takes values from 0 to 2.
Therefore, the right-hand limit is 0.
Since the left-hand limit and right-hand limit are not equal, the limit does not exist.
Hence, the correct option is none of the above.
Note: The greatest integer function cannot be applied to a limit. The limit should be evaluated first and then the greatest integer function can be applied to the result.
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The value of lim x->0 [sinx/x], where [.] represents the greatest inte...
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The value of lim x->0 [sinx/x], where [.] represents the greatest integer function is a) 1 b) 0 c) -1 d) none of the above The correct answer is option 'b', can you please explain this answer?
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