undersetx→ π /2 mathop lim ( sin x/ cos -1[ (1)4(3 sin x- sin 3x) ]), where [.] denotes greatest integer function, is

Question

undersetx→ π /2 mathop lim ( sin x/ cos -1[ (1)4(3 sin x- sin 3x) ]), where [.] denotes greatest integer function, is (A)  (2/π )  (B) 1 (C)  (4/π )  (D) Does not exist

Tardigrade Admin · Accepted Answer

Given limit is L= undersetx→ (π /2) mathop lim ( sin x/ cos -1[ sin 3x]) Now, when x→ (π /2),[ sin 3x]→ 0 as sin x→ 1 ∴ L=(2/π )

Q. $x \to π /2 lim \frac{s i n x}{c o s ^{- 1} [ \frac{1}{4} ( 3 s i n x - s i n 3 x ) ]},$ where $[.]$ denotes greatest integer function, is

$\frac{2}{π}$

1

$\frac{4}{π}$

Does not exist

Given limit is $L = x \to \frac{π}{2} lim \frac{s i n x}{c o s ^{- 1} [ s i n ^{3} x ]}$ Now, when $x \to \frac{π}{2}, [sin^{3} x] \to 0$ as $sin x \to 1$ $∴$ $L = \frac{2}{π}$

Q. x→π/2lim​cos−1[41​(3sinx−sin3x)]sinx​, where [.] denotes greatest integer function, is

π2​

1

π4​

Does not exist

Given limit is L=x→2π​lim​cos−1[sin3x]sinx​ Now, when x→2π​,[sin3x]→0 as sinx→1 ∴ L=π2​

Q. $x \to π /2 lim \frac{s i n x}{c o s ^{- 1} [ \frac{1}{4} ( 3 s i n x - s i n 3 x ) ]},$ where $[.]$ denotes greatest integer function, is

$\frac{2}{π}$

$\frac{4}{π}$

Given limit is $L = x \to \frac{π}{2} lim \frac{s i n x}{c o s ^{- 1} [ s i n ^{3} x ]}$ Now, when $x \to \frac{π}{2}, [sin^{3} x] \to 0$ as $sin x \to 1$ $∴$ $L = \frac{2}{π}$