What is `lim_(xto0) ((1+x)^(n)-1)/(x)` equal to ?

Options

A. 0
B. 1
C. n
D. n-1

Answer

C

Explanation

`underset(xto0)lim((1+x)^(n)-1)/(x)`
`=underset(xto0)lim(""^nC_(0)+""^(n)C_(1)x+""^(n)C_(2)x^(2)+......+""^(n)C_(n)x^(n)-1)/(x)`
`=underset(xto0)lim(x(""^(n)C_(1)+""^(n)C_(2)x+......+""^(n)C_(n)x^(n-1)))/(x)`
`=underset(xto0)lim""^(n)C_(1)+""^(n)C_(2)x+......+""^(n)C_(n)x^(n-1)`
Put `x=0implies""^(n)C_(1)=n`