We must to calculate lim x → π / 2 sin ⁡ x − 1...

Boilanubjaini8f

Boilanubjaini8f

Answered

2022-06-28

We must to calculate
lim x π / 2 sin x 1 2 x π .

Answer & Explanation

Zayden Wiley

Zayden Wiley

Expert

2022-06-29Added 21 answers

We must to calculate
lim x π / 2 sin x 1 2 x π
If we use a change of variables   h = x π 2 ,   then this becomes:
lim h   0 sin ( π 2 + h ) sin ( π 2 ) 2 h
= ( ) 1 2   f ( π 2 ) ,
where   f ( x ) = sin ( x ) .   All we have done at (*) is use the definition of   f ( x ) .
But   f ( π 2 ) = 0 ,   and therefore the answer is   1 2 × 0 = 0.
Also, from the comments, you went wrong here:
lim t 0 sin ( t + π 2 ) sin π 2 2 t = lim t 0 2 sin ( t + π ) cos t 2 t
This is incorrect. The correct formula is:
  sin α sin β = 2 cos α + β 2 sin α β 2 ,   which would get you:
  sin ( t + π 2 ) sin ( π 2 ) = 2 cos ( 1 2 ( t + π ) ) sin ( 1 2 t )
Yesenia Sherman

Yesenia Sherman

Expert

2022-06-30Added 5 answers

Another way to rewrite it is
sin ( t + π / 2 ) 1 2 t = sin t cos ( π / 2 ) + sin ( π / 2 ) cos t 1 2 t = cos t 1 2 t .

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