How do I find a sequence that satisfies this limit?

Question

I have that $ n(1 - (1 - \frac{1}{x})^3) \rightarrow \theta $ as $n \rightarrow \infty $ where $\theta$ is a constant. I am trying to find $x$ that satisfies this limit. After expanding the LHS I still cannot find any solutions for $x$. Any ideas would be greatly appreciated.

Answer 1

There is no such $x$. For a much simpler case, consider you want to find $y$ such that $\lim_{n \to \infty}ny=\theta$. If $y \gt 0,$ the limit is $+\infty$. If $y=0,$ the limit is $0$. If $y \lt 0,$ the limit is $-\infty$

Answer 2

Subbing in $x = \frac{3n}{\theta}$, your expression expands as

\begin{align*} n\left(1-\left(1-\frac{\theta}{3n}\right)^3\right) &= n\left(1-\left(1-3\cdot\frac{\theta}{3n} + 3\cdot\frac{\theta ^2}{9n^2} - \frac{\theta ^3}{27n^3}\right)\right)\\ &=\theta - 3\cdot\frac{\theta^2}{9n} + \frac{\theta^3}{27n^2}. \end{align*}

This gives the limit that you want.