1.

Let f(x)={(0 , x "is irrational"),(2/(2q^(3)-q^(2)+q+sin^(2)q+5) , if x=p/q ("rational")):}(where HCF (p,q)=1,p,q,gt0) and f(x) is defined AAxgt0 then which of the following is/are incorrect?

Answer»

`f(x)` is continuous at each irrational in `(0,oo)`
`f(x)` is continuous at each rational in `(0,oo)`
`f(x)` is discontinuous at each rational in `(0,oo)`
`f(x)` is discontinuous for all `x` in `(0,oo)`

Solution :LET `x=SQRT(3)`
`f(sqrt(3))=0`
`because sqrt(3)=1.732050807`……….
As the decimal PART increase then in the expression `p/q, q` becomes very large
So, `2/(2q^(3)-q^(2)+q+sin^(2)q+t)to0`
Hence `lim_(xtosqrt(3))f(x)=0`
THUS, `f(x)` is continuous at each irrational


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