# Evaluating lim_(n => infty) (1-(x)/(n^(1+a)))^n

Garrett Sheppard 2022-08-14 Answered
Evaluating $\underset{n\to \mathrm{\infty }}{lim}{\left(1-\frac{x}{{n}^{1+a}}\right)}^{n}$
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Brennan Parks
That step you are uncomfortable with is indeed quite dodgy. Instead, consider the logarithm of your limit. Since $\mathrm{log}\left(1-t\right)=-t+{t}^{2}/2+{t}^{3}/3+\cdots ,$
$n\mathrm{log}\left(1-\frac{x}{{n}^{1+a}}\right)=-\left(\frac{x}{{n}^{a}}+\frac{{x}^{2}}{2{n}^{1+2a}}+\frac{{x}^{3}}{3{n}^{2+3a}}+\cdots \right)$
so as $n\to \mathrm{\infty },$ if $a=0$ this goes to $-x$, if $a>0$ to goes to 0 and if $a<0$ this goes to $-\mathrm{\infty }.$. Exponentiating gives back the result.