Approximation of $(1 - 1/n)^m$ when $m$ depends on $n$

Question

Specifically, is it correct to approximate the expression $\left(1 - 2^{-t}\right)^{x \cdot 2^t}$ as $e^{-x}$? This answer from an older post suggests that for an expression of the form $(1 - 1/n)^m$, quote

If $m$ depends of $n$ or the expression is part of a bigger term, it must be considered as a whole.

and so

[...] the sequence converges to 1.

which would contradict the approximation $e^{-x}$ of the original expression.

Answer 1

We have that for any $f(t) \to \infty$

$$\left(1-\frac1{f(t)}\right)^{x\cdot f(t)} \to e^{-x}$$

which is related and can be proved by the standard limit

$$\left(1+\frac1{n}\right)^{n} \to e$$

The case $(1 - 1/n)^m$ with $n\to \infty$ and $m\to \infty$ is undetermined if we don't have any information about the rate of growing of $m$ with respect to $n$, indeed for example

• $m=kn$

$$\left(1+\frac1n\right)^{kn} \to e^k$$

• $m=n^2$

$$\left(1+\frac1n\right)^{n^2} \to \infty$$