If `m , n in N ,("lim")_(xvec0)(sinx^m)/((sinx)^m)i s` `1,ifn=m` (b) `0,ifn > m` `oo,ifn

Options

A. `(p)/(2)`
B. `(pi)/(2sqrt(2))`
C. `(pi)/(sqrt(2))`
D. `sqrt(2)pi`

Answer

A, B, C

Explanation

`L=underset(xto0)lim(sinx^(n))/((sinx)^(m))=underset(xto0)lim((sinx^(n))/(x^(n))x^(n))/(((sinx)^(m))/(x^(m))x^(m))=underset(xto0)limx^(n-m)`
If `n=m`, then
`L=("a very small value near to zero")^("exactly zero")=1`
If `ngtm`, then
`L=("a very small value near to zero")^("Positive integer")=0`
If `nltm`, then
`L=("a very small value near to zero")^("negative integer")=oo`