Question Number 101148 by john santu last updated on 30/Jun/20

lim_(x→0) ((sin (ln (1+x))−ln(1+sin x))/(sin ^4 ((x/2)))) =?

Commented bybramlex last updated on 01/Jul/20

lim_(x→0) ((((cos (ln(1+x)))/(1+x)) − ((cos x)/(1+sin x)))/((1/2)×4sin ^3 ((x/2))×cos ((x/2))))  (1/2)×lim_(x→0) ((((cos (ln(1+x)))/(1+x)) − ((cos x)/(1+sin x)))/(sin ^3 ((x/2))))  continue

Commented byDwaipayan Shikari last updated on 01/Jul/20

lim_(x→0) ((sin(log(1+x)))/(log(1+x))) .((log(1+x)cos^4 (x/2))/(sin^4 ((x/2))cos^4 (x/2)))−((log(1+sinx))/(sin^4 (x/2)cos^4 (x/2))) .cos^4 (x/2)  lim_(x→0)  16((log(1+x))/(sin^4 x))−((16)/(sin^3 x))  lim_(x→0) 16((log(1+x))/x^4 )−((16)/x^3 )=lim_(x→0)    ((16)/x^3 )−((16)/x^3 )=0    {As x→0 so sin^4 x=x^4 {(((sin(log(1+x)))/(log(1+x)))=1}    as  x→0  ((log(1+x))/x)=1