Find the limits of the following: `(i)lim_(xto0) (sin3x)/(x)" "(ii)lim_(xto0) (sin7x)/(sin4x)" "(iii)lim_(xto0) (1-cos^(2)x)/(x^(2))`

Explanation

`(i)underset(xto0)lim(sin3x)/(x)=3underset(xto0)lim(sin3x)/(3x)`
`3xx1["as " underset(xto0)lim(sin(f(x)))/(f(x))=1, if underset(xto0)limf(x)=0]`
=3
`(ii)underset(xto0)lim(sin7x)/(sin4x)=underset(xto0)lim((sin7x)/(7x))/((sin4x)/(4x))xx(7x)/(4x)`
`(underset(xto0)lim((sin7x))/(7x))/(underset(xto0)lim(sin4x)/(4x))xx7/4`
`=1/1xx7/4=7/4`
`(iii
)underset(xto0)lim(1-cos2x)/(x^(2))=underset(xto0)lim(2sin^(2)x)/(x^(2))`
`=2(underset(xto0)lim(sinx)/(x))^(2)=2`