Question

lim x->0 ( 1 - x )^n - 1 / x​

Answer 1

Step-by-step explanation:

$\tt \lim \limits _{x\to0} \dfrac{(1 - x) {}^{n } - 1 }{x}$

Answer :

Put 1- x=y

$\color{darkcyan}\[ \begin{aligned} \tt \text { As } x \rightarrow 0 \Rightarrow y \rightarrow 1 & \\ \\ \tt\therefore \quad \lim _{x \rightarrow 0} \frac{(1-x)^{n}-1}{x} & \tt=\lim _{y \rightarrow 1} \frac{y^{n}-1}{1-y} \\ \\ & \tt=-\lim _{y \rightarrow 1} \frac{y^{n}-1^{n}}{y-1} \\ \\ & \tt=-n\left(1^{n}-1\right) \\ \\ & \tt=-n . \end{aligned} \]$