`("lim")_(xto0)((2^m+x)^(1/m)-(2^n+x)^(1/n))/x is equal t o` (a) 2`(1/(m2^m)-1/(n2^n))' (b) `(1/(m2^m)+1/(n2^n))` (c)`1/(m2^(-m))-1/(n2^(-n))` (d) `1/(m2^(-m))+1/(n2^(-n))`

Options

A. `(n!)^(n)`
B. `(n!)^(1//n)`
C. `n!`
D. `ln(n!)`

Answer

C

Explanation

`underset(xto0)lim((2^(m)+x)^(1//m)-(2^(n)+x)^(1//n))/(x)`
`underset(xto0)lim((2^(m)+x)^(1//m)-2)/(x)-underset(xto0)lim((2^(n)+x)^(1//n)-2)/(x)`
`=underset(ato2)lim(a-2)/(a^(m)-2^(m))-underset(bto0)lim(b-2)/(b^(n)-2^(n))`
`["Putting "2^(m)+x=a^(m)" and "2^(n)+x=b^(n)]`
`=(1)/(m2^(m-1))-(1)/(n2^(n-1))`