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`underset(xrarr0^(+))limf'((1)/(x))=1``underset(xrarr0^(+))limxf((1)/(x))=2``underset(xrarr0^(+))limx^(2)f'(x)=0``|f(x)|le2" for all "x in (0,2)`

Solution :

`f'(x)+(f(x))/(x)=2` <br> `rArr" "xf'(x)+f(x)=2x` <br> `rArr" "int d(x.f(x))=int 2xdx` <br> `rArr" "xf(x)=x+(c)/(x)" "(c ne 0 as f(1) ne1)` <br> `underset(xrarr0^(+))limf'((1)/(x))=underset(xrarr0^(+))lim(1-cx^(2))=1` <br> `underset(xrarr0^(+))limxf((1)/(x))=underset(xrarr0^(+))lim(1+cx^(2))=1` <br> `underset(xrarr0^(+))limx^(2)f'(x)=underset(xrarr0^(+))lim(x^(2)-c)=-c ne 0` <br> `underset(xrarr0^(+))limf(x)=oo or -oo" so option (4) is incorrect "` **Recall Trigonometric Functions**

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