Question

Evaluate $\underset{x\to 0}{lim}\frac{{(1-x)}^n-1}{x}$

Answer 1

Given

$\underset{x\to 0}{lim}\left(\frac{(1-x{)}^n-1}{x}\right)$

Applying L'Hopital's Rule

$=\underset{x\to 0}{lim}\left(\frac{-n(-x+1{)}^{n-1}}{1}\right)$

$=\underset{x\to 0}{lim}(-n(-x+1{)}^{n-1})$

Putting the value x=0

$=-n(0+1{)}^{n-1}$

$=-n\left(1^{n-1}\right)$