(i) Let `h (x) = underset(x to oo)lim(x^(2n) f(x) + g(x))/(1+x^(2n))`, find h(x) in terms of f(fx) and g(x)
(ii) without using L Hospital rule or series expansion for `e^(x) `evaluate ` underset(x to 0) lim (e^(x) -1-x)/x^(2)`
(iii) ` underset(n to oo) lim [ e^(1/n)/n^(2) + 2 ((e^(1/n))^(2))/n^(2) + 3. ((e^(1/n))^(3))/n^(2)+.......+ n((e^(1/n))^(n))/n^(2)]`
(iv) `underset(x to 0)lim[ (a sin x)/x ] + [ (b tan x)/x]` Where a,b are inegers and [] denotes integral part.
(v) `underset(x to a)lim (sinx/sina)^(1/(x-a))`

Explanation

Let's solve the given problems step by step.

### Part (i)
We need to find \( h(x) = \lim_{x \to \infty} \frac{x^{2n} f(x) + g(x)}{1 + x^{2n}} \).

1. **Case 1**: When \( x^{2n} \) is less than 1 (which is not possible as \( x \to \infty \)).
- This case does not apply as \( x^{2n} \) will always be greater than 1 for large \( x \).

...