Evaluate the following limits....fast reply appreciated its urgent?

Question

1) lim x--->pi sin x / pi - x
2) lim x---> pi/2 sin 2x / cos x
3) lim x---> pi/2 cos^2 x / 1-sin x
plzz explain step by step in detail thanks but not by hopitals rule

Answer 1

2). lim x---> pi/2 sin 2x / cos x

2

There fore

 

3). lim x---> pi/2 cos^2 x / 1-sin x

There fore

Answer 2

1)

Therefore lim x--->pi sin (x / pi - x) = 1.

Comments

lim x--->pi sin (x) / (pi - x)

 

sinx can be written as sinx =-sin (x-pi)

 

lim x-pi--->0 -sin (x-pi) / -(x - pi)

 

lim x-pi--->0 sin (x-pi) / (x - pi)

 

Use the following formula lim θ--->0 sin (θ) / (θ)=1

 

lim x-pi--->0 sin (x-pi) / (x - pi)=1