If `l =lim_(xrarr0) (tanx^(n))/((tanx)^m)`, where `m,n in N`, then

Options

A. `l=1 " for all " m,n in N`
B. `l={(1" if "n gt m,,),(0" if "n lt m,,):}`
C. `l={(1" if "n = m,,),(0" if "n gt m,,):}`
D. `l=0 " for all " m,n in N`

Answer

C

Explanation

We have,
` l=lim_(xto0) (tanx^n)/((tanx)^m)`
` rArr l=lim_(xto0) ((tanx^n)/(x^n))xx(x^n)/(x^m)xx((x)/(tanx))^m`
` rArr l=lim_(xto0) 1xx x^(n-m)xx1`
`rArr l=lim_(xto0) x^(n-m)rArr l={(0,if,ngtm,),(1,if,n=m,),(oo,if,n=m,):}`