Question

If `m, n in I_(0)` and `lim_(xto0) (tan2x-nsinx)/(x^(3))` = some integer, then find the value of n and also the value of limit.

Answer 1

`L=underset(xto0)lim(tan2x-nsinx)/(x^(3))`
`underset(xto0)lim(sin2x-nsinxcos2x)/(x^(3)cos2x)`
`=underset(xto0)lim("sin"x)/(x)((2cosx-ncos2x))/(x^(2))=(1)/(cos2x)`
`=underset(xto0)lim((2cosx-ncos2x))/(x^(2))`
Now, for `xto0`,`x^(2)to0`.
Therefore, for `xto0, 2cosx-ncos2xto0.` So, n=2. For, n=2.
`L=underset(xto0)lim((2cosx-ncos2x))/(x^(2))`
`=4underset(xto0)lim("sin"(x)/(2)"sin"(3x)/(2))/(x^(2))`
=3