Friend , Please elaborate your answer . What do you mean by the convergence of limit ?.

Ques. Does the series expansion in limits chapter is only applied when the variable ( x ) tends to zero or is small? can we use it in general?

In accordance to the second Ques.

Friend , I am not satisfied with your answer . What you are saying that the given function is an oscillating one but i reall disagree that when {x}=0 then the given function would be zero . Actually it is assuming 0/0 indeterminate form.

Ques.   {(x/x+1)^a + sin(1/x)}^x

Solution given: e^1-a

My solution : e

Ques . {x} denotes the fractional part of x. then   ({x}/tan{x})

solution given : not in the option (none of these )

my solution :1

Ques. if f(x)=1/3{f(x+1)+[5/f(x+2)]} and f(x)>0 for all  x belongs to R ,  then f(x)

Solution given : (5/2)^1/2

^{Specially do the third Question ,}

Thanks , Allam, you have solved my problem .

Even i am expecting the same answer but the book says something else . I know the book answer can be wrong but its probability is quite low . There should be some other logic , some other way to do this question .

I am sorry my friend , but the answer is (1-x)/x

you can find this question in Objective Approach -Mathematics by Arihant Publication , Work Book Exercise 7.2

        Ques.  if  x^{e^y+e^y.......infinity} 

               then find dy/dx

Note: e^y+e^y....  means e^{y+e^y......infinity}