A right-angled triangle has one leg of length 1 unit, another leg of length x units and hypotenuse is of length y units. The angle opposite to the side of length x units is $θ$ then:

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$\lim_{θ \to \frac{π}{2}} (y - x) = 0$

$y^{3} - x^{3} = \sec^{3} θ - \tan^{3} θ$

$\lim_{θ \to \frac{π}{2}} (y^{3} - x^{3}) = - 1$

$\lim_{θ \to \frac{π}{2}} (y^{2} - x^{2}) = 1$

answer is A, B, C.

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Detailed Solution

$\begin{array}{l} x = \tan θ, y = \sec θ \\ (A) y^{3} - x^{3} = \sec^{3} θ - \tan^{3} θ \\ (B) \lim_{θ \to \frac{π}{2}} (\sec θ - \tan θ) = \lim_{θ \to \frac{π}{2}} \frac{1 - \sin θ}{\cos θ} = 0 \\ (C) \lim_{θ \to \frac{π}{2}} (\sec^{2} θ - \tan^{2} θ) = 1 \\ (D) \lim_{θ \to \frac{π}{2}} (\sec^{3} θ - \tan^{3} θ) = D . N . E \end{array}$ Question Image