displaystyle limx → (π/2) ( cot x - cos x/(π - 2x)3) equals :

Question

displaystyle limx → (π/2) ( cot x - cos x/(π - 2x)3) equals : (A) (1/16) (B) (1/8) (C) (1/4) (D) (1/24)

Tardigrade Admin · Accepted Answer

displaystyle limx → (π/2) ( cot x - cos x/(π - 2x)3) Put, (π/2)-x = t displaystyle limt → 0( tan t- sin t/8t3) = displaystyle limt → 0( sin t⋅ 2 sin2 (t/2)/8t3) = (1/16).

Q. $x \to \frac{π}{2} lim \frac{cot x - cos x}{( π - 2 x ) ^{3}}$ equals :

$\frac{1}{16}$

$\frac{1}{8}$

$\frac{1}{4}$

$\frac{1}{24}$

$x \to \frac{π}{2} lim \frac{cot x - cos x}{( π - 2 x ) ^{3}}$
Put, $\frac{π}{2} - x = t$
$t \to 0 lim \frac{tan t - sin t}{8 t ^{3}}$
$= t \to 0 lim \frac{sin t \cdot 2 sin ^{2} \frac{t}{2}}{8 t ^{3}}$
$= \frac{1}{16} .$

Q. x→2π​lim​(π−2x)3cotx−cosx​ equals :

161​

81​

41​

241​