please help to find out the limit as n-tends to > infinity for the (e) option here

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You asked for part(e) but your arrow points to part (d). Anyway, here is part (e).

Divide all 4 terms by sqrt(n). This will get you:

{ (n + 1/n)^.5 + 1} / (n + 1/n)^.25 - 1}

It's all over now as the numerator takes off faster than the denominator. So the expression ----> infinity.

If that's not enough rigour for you then let m =( n + 1/n)^.25

As n ---> infinity then m also ---> infinity

So now we have: { m^2 + 1} /{ m-1} as m ---> infinity and you can take it from there.