What is equal ? #lim_(y->x)(sin^2y -sin^2x )/(y^2-x^2)= #

2 Answers
maganbhai P.
Mar 5, 2018

#(sin2x)/(2x)#

Explanation:

#color(blue)(sin^2A-sin^2B=sin(A-B)*sin(A+B))#
#L=lim_(y->x)(sin^2y-sin^2x)/(y^2-x^2),#
We know that , #color(red)(y to xrArr(y-x) to 0)#
#:.L=lim_((y-x)->0)(sin(y-x)sin(y+x))/((y-x)(y+x))#
#:.L=lim_((y-x)-.0)(sin(y-x))/(y-x)*lim_(y->x)(sin(y+x))/(y+x)#
#:.L=(1)*(sin(x+x))/(x+x),as,color(red)( lim_(theta->0)(sintheta)/(theta)=1,theta=(y-x))#
#:.L=(sin2x)/(2x)#

Ratnaker Mehta
Mar 5, 2018

# (sin2x)/(2x)#.

Explanation:

Reall that, #sin^2A-sin^2B=sin(A+B)sin(A-B)#.

#:."The Reqd. Lim.="lim_(y to x)(sin^2y-sin^2x)/(y^2-x^2)#,

#=lim_(y to x){sin(y+x)*sin(y-x)}/{(y+x)(y-x)#,

#=lim_(y to x)sin(y-x)/(y-x)*sin(y+x)/(y+x)#,

#=lim_((y-x) to 0)sin(y-x)/(y-x)*lim_(y to x)sin(y+x)/(y+x)#,

#=1*sin(x+x)/(x+x)#,

#=(sin2x)/(2x)#.

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