`lim_(xto0) ((2^(m)+x)^(1//m)-(2^(n)+x)^(1//n))/(x)` is equal to

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))`