The value of displaystyle lim x arrow ∞ arc cos ((arc sin ((π/x))+arc tan ((x/π))/x sin (π)x)) is equal to

Question

The value of displaystyle lim x arrow ∞ arc cos ((arc sin ((π/x))+arc tan ((x/π))/x sin (π)x)) is equal to (A) (π /3) (B) (π /6) (C) (π /4) (D) (π /2)

Tardigrade Admin · Accepted Answer

cos- 1(((sin)- 1 (π /x) + (tan)- 1 (x/π )/x ((sin (π )x/((π )x))) ⋅ (π )x) =cos/- 1)((0 + (π /2)/1 ⋅ π ))=cos- 1((1/2))=(π /3) .

Q. The value of $x \to \infty lim a rc cos (\frac{a rc sin ( \frac{π}{x} ) + a rc tan ( \frac{x}{π} )}{x sin \frac{π}{x}})$ is equal to

$\frac{π}{3}$

$\frac{π}{6}$

$\frac{π}{4}$

$\frac{π}{2}$

$co s^{- 1} ⎝ ⎛ \frac{( s in ) ^{- 1} \frac{π}{x} + ( t an ) ^{- 1} \frac{x}{π}}{x ( \frac{s in \frac{π}{x}}{( \frac{π}{x} )} ) \cdot \frac{π}{x}} ⎠ ⎞$
$= co s^{- 1} (\frac{0 + \frac{π}{2}}{1 \cdot π}) = co s^{- 1} (\frac{1}{2}) = \frac{π}{3}$ .

Q. The value of x→∞lim​arccos(xsinxπ​arcsin(xπ​)+arctan(πx​)​) is equal to

3π​

6π​

4π​

2π​