Binomial-type and power limits for JEE

Formula

\[\lim_{x\to 0}\frac{(1+x)^n-1}{x}=n,\qquad \lim_{x\to 0}\frac{(1+x)^\alpha-1}{x}=\alpha\]

Variables: \(n\in\mathbb N\), \(\alpha\in\mathbb R\).

Conditions: For real \(\alpha\), require \(1+x>0\) near \(0\).

Simplification of algebraic and exponential limits.