displaystyle limx → (π /2) (1-cos x/sin x) is

Question

displaystyle limx → (π /2) (1-cos x/sin x) is (A) 0 (B) -1 (C) 1 (D) Does not exist

Tardigrade Admin · Accepted Answer

We have, displaystyle limx → (π /2) (1-cos x/sin x) = displaystyle limx → (π /2) (2 sin2 ((x/2))/2 sin ((x)2)cos((x)2) = displaystyle lim/x → (π )2)tan ((x/2)) =tan ((π/4))=1

Q. $x \to \frac{π}{2} lim$ $\frac{1 - cos x}{s in x}$ is

$0$

$- 1$

$1$

Does not exist

We have, $x \to \frac{π}{2} lim$ $\frac{1 - cos x}{s in x}$
$= x \to \frac{π}{2} lim$ $\frac{2 s i n ^{2} ( \frac{x}{2} )}{2 s in ( \frac{x}{2} ) cos ( \frac{x}{2} )}$
$= x \to \frac{π}{2} lim$ $t an (\frac{x}{2})$
$= t an (\frac{π}{4}) = 1$

Q. x→2π​lim​ sinx1−cosx​ is

0

−1

1

Does not exist

We have, x→2π​lim​ sinx1−cosx​ =x→2π​lim​ 2sin(2x​)cos(2x​)2sin2(2x​)​ =x→2π​lim​tan(2x​) =tan(4π​)=1

Q. $x \to \frac{π}{2} lim$ $\frac{1 - cos x}{s in x}$ is

$0$

$- 1$

$1$

We have, $x \to \frac{π}{2} lim$ $\frac{1 - cos x}{s in x}$
$= x \to \frac{π}{2} lim$ $\frac{2 s i n ^{2} ( \frac{x}{2} )}{2 s in ( \frac{x}{2} ) cos ( \frac{x}{2} )}$
$= x \to \frac{π}{2} lim$ $t an (\frac{x}{2})$
$= t an (\frac{π}{4}) = 1$