The value of undersetx arrow 0l i m (1 - c o s3 (sin x)/sin â¡ x s i n (sin â¡ x) c o s (sin â¡ x)) is equal to

Question

The value of undersetx arrow 0l i m (1 - c o s3 (sin x)/sin â¡ x s i n (sin â¡ x) c o s (sin â¡ x)) is equal to (A) (3/2) (B) 1 (C) 0 (D) 2

Tardigrade Admin · Accepted Answer

Let, sin x=t ⇒ undersett arrow 0l i m(1 - c o s3 t/t sin t cos â¡ t) ⇒ undersett arrow 0l i m((1 - cos t)/t2)× ((t/sin â¡ t))× ((1 + cos â¡ t + (cos)2 â¡ t)/cos â¡ t) ⇒ (1/2)× 1× ((1 + 1 + 1)/1)=(3/2)

Q. The value of $x \to 0 l im \frac{1 - co s ^{3} ( s in x )}{s in x s in ( s in x ) cos ( s in x )}$

is equal to

$\frac{3}{2}$

$1$

$0$

$2$

Let, $s in x = t$
$\Rightarrow t \to 0 l im \frac{1 - co s ^{3} t}{t s in t cos t}$
$\Rightarrow t \to 0 l im \frac{( 1 - cos t )}{t ^{2}} \times (\frac{t}{s in t}) \times \frac{( 1 + cos t + ( cos ) ^{2} t )}{cos t}$
$\Rightarrow \frac{1}{2} \times 1 \times \frac{( 1 + 1 + 1 )}{1} = \frac{3}{2}$

Q. The value of x→0lim​sin⁡xsin(sin⁡x)cos(sin⁡x)1−cos3(sinx)​ is equal to

23​

1

0

2

Let, sinx=t ⇒t→0lim​tsintcos⁡t1−cos3t​ ⇒t→0lim​t2(1−cost)​×(sin⁡tt​)×cos⁡t(1+cos⁡t+(cos)2⁡t)​ ⇒21​×1×1(1+1+1)​=23​

Q. The value of $x \to 0 l im \frac{1 - co s ^{3} ( s in x )}{s in x s in ( s in x ) cos ( s in x )}$

is equal to

$\frac{3}{2}$

$1$

$0$

$2$

Let, $s in x = t$
$\Rightarrow t \to 0 l im \frac{1 - co s ^{3} t}{t s in t cos t}$
$\Rightarrow t \to 0 l im \frac{( 1 - cos t )}{t ^{2}} \times (\frac{t}{s in t}) \times \frac{( 1 + cos t + ( cos ) ^{2} t )}{cos t}$
$\Rightarrow \frac{1}{2} \times 1 \times \frac{( 1 + 1 + 1 )}{1} = \frac{3}{2}$