Evaluate `(i)lim_(xrarr0)((1-cos 4x)/(1-cos5x))`
`(ii) lim_(xrarr0)((1-cosmx)/(1-cosnx)).`

Explanation

`(i)lim_(xto0)((1-cos4x)/(1-cos5x))=lim_(xto0)(2sin^(2)2x)/(2sin^(2)(5x//2))`
`=lim_(xto0)(sin^(2)2x)/(sin^(2)(5xx//2))=lim_(xto0){({(sin2x)/(2x)}^(2).4x^(2))/({(sin(5x//2))/((5x//2))}^(2).(25x^(2))/(4))}`
`16/25.(lim_(2xto0)((sin2x)/(2x))^(2))/(lim_((5x)/(x)to0){(sin(5x//2))/((5x//2))}^(2))=((16)/(25)xx(1^(2))/(1^(2)))=16/25.`
`(ii)lim_(xto0)((1-cosmx)/(1-cosnx))`
`=lim_(xto0){(2sin^(2)(mx//s))/(2sin(nx//2))}=lim_(xto0){(sin^(2)(mx//2))/(sin^(2)(nx//2))}`
`=lim_(xto0){((sin^(2)(mx//2))/((mx//2)^(2))xx(m^(2)x^(2))/(4))/((sin^(2)(nx//2))/((nx//2)^(2))xx(n^(2)x^(2))/(4))}=(m^(2))/(n^(2)).(lim_((mx)/(2)to0){(sin(mx//2))/((mx//2))}^(2))/(lim_((nx)/(2)to0){(sin(nx//2))/((nx//2))}^(2))`
`(m^(2)/(n^(2))xx(1^(2))/(1^(2)))=(m^(2))/(n^(2)).`