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Question

If {x} and [x] are the fractional part function and greatest integer functions of x respectively, then limx[a]e{x}{x}1{x}2


Your Answer
A
Your Answer
B
Your Answer
C
Correct Answer
D
Does not exist

Solution

The correct option is D Does not exist
LHL=limx[a]e{x}{x}1{x}2=limh0e{[a]h}{[a]h}1{[a]h}2=limh0e1h(1h)1(1h)2=e2RHL=limx[a]+e{x}{x}1{x}2=limh0e{[a]+h}{[a]+h}1{[a]+h}2=limh0ehh1h2=P(say)Replacing h by h in P, thenP=limh0eh+h1h2Adding Eq.(i) and (ii), then2P=limh0eh+eh2h2=limh0(eh1)2eh.h2=(1)21=1P=12RHL=12     [from Eq.(i)]Hence, LHLRHL
 

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