Evaluate the following limits
(i) `lim_(x to (pi)/(2)) tan^(2) x [sqrt(2 sin^(2) x + 3 sin x + 4) - sqrt(sin^(2) x + 6 sin x + 2)]`
(ii) `lim_(theta to 0) (sqrt(1 + sin 3 theta) -1)/(ln(1 + tan 2theta))`
(iii) `lim_(x to 0) (sqrt(1 + x) - ""^(3)sqrt(1 + x))/(x)`
(iv) `lim_(phi to 0) (8)/(phi^(8)) (1 - cos (phi^(2))/(2) - cos (phi^(2))/(4) + cos (phi^(2))/(2). cos (phi^(2))/(4))`

Explanation

Let's evaluate the limits step by step.

### (i) Evaluate \( \lim_{x \to \frac{\pi}{2}} \tan^2 x \left( \sqrt{2 \sin^2 x + 3 \sin x + 4} - \sqrt{\sin^2 x + 6 \sin x + 2} \right) \)

1. **Substitution**: As \( x \to \frac{\pi}{2} \), \( \tan^2 x \) approaches infinity and both square root expressions approach \( \sqrt{4} = 2 \). This gives us an indeterminate form \( \infty \cdot 0 \).

2. **Rationalization**: We can rationalize the expression:
\[
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