If where p, q, r and t ∈ R and n ∈ N, thenList-IList-II(P) the value of q is (1) 0(R) ther value of n is (2) 2(Q) the value of (q + r) is(3) 4(S) Thye vale of (r + t) is (4) 6

Engineering

Mathematics

Trigonometric Limits

Question

If $\underset{x \to 0}{Lim} \frac{\tan^{2} (tanx) - \tan^{2} x}{{px}^{6} + {qx}^{n} + rx + t} = \frac{1}{9}$ where p, q, r and t ∈ R and n ∈ N, then

P → 3

Q → 1

Q → 4

S → 1

P → 4

Q → 3

Q → 2

S → 1

P → 2

Q → 1

Q → 2

S → 3

P → 4

Q → 3

Q → 4

S → 1

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$\underset{x \to 0}{Lim} \frac{(\tan (tanx) + tanx) (\tan (tanx) - tanx)}{{px}^{6} + {qx}^{n} + rx + t} = \frac{1}{9}$

t must be zero

$\underset{x \to 0}{Lim} (\frac{\tan (tanx)}{tanx \cdot \frac{x}{tanx}} + \frac{tanx}{x})$

$\frac{\sin (tanx - x)}{\cos (tanx) cosx ({px}^{5} + {qx}^{n - 1} + r)} = \frac{1}{9}$

$2 \cdot \underset{x \to 0}{Lim} (\frac{tanx - x}{x^{3}}) (\frac{x^{3}}{{px}^{5} + {qx}^{n - 1} + r}) = \frac{1}{9}$

r = 0, n – 1 = 3

⇒ n = 4 and p ∈ R

$∴ 2 \times \frac{1}{3} \times \frac{1}{q} = \frac{1}{9}$

⇒ q = 6