Question

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

Answer 1

`(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)`

Answer 2

`(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)`.