If displaystyle lim x arrow 0[1+x ln (1+b2)]1 / x=2 b sin 2 θ, b0 and θ ∈(-π, π], then the value of θ is

Question

If displaystyle lim x arrow 0[1+x ln (1+b2)]1 / x=2 b sin 2 θ, b>0 and θ ∈(-π, π], then the value of θ is (A) ± (π/4) (B) ± (π/3) (C) ± (π/6) (D) ± (π/2)

Tardigrade Admin · Accepted Answer

e ln (1+ b 2)=2 b sin 2 θ ⇒ sin 2 θ=(1+ b 2/2 b ) ⇒ sin 2 θ=1 as (1+ b 2/2 b ) ≥ 1 θ=± π / 2

Q. If $x \to 0 lim [1 + x ln (1 + b^{2})]^{1/ x} = 2 b sin^{2} θ, b > 0$ and $θ \in (- π, π]$ , then the value of $θ$ is

$\pm \frac{π}{4}$

$\pm \frac{π}{3}$

$\pm \frac{π}{6}$

$\pm \frac{π}{2}$

$e^{l n (1 + b^{2})} = 2 b sin^{2} θ$
$\Rightarrow sin^{2} θ = \frac{1 + b ^{2}}{2 b}$
$\Rightarrow sin^{2} θ = 1$ as $\frac{1 + b ^{2}}{2 b} \geq 1$
$θ = \pm π /2$

Q. If x→0lim​[1+xln(1+b2)]1/x=2bsin2θ,b>0 and θ∈(−π,π], then the value of θ is

±4π​

±3π​

±6π​

±2π​