I am having a huge brain fart while helping a 12 year old student at Mathnasium on this:

using the fact that

$$\lim_{n\to 0}(1+n)^{x/n} = \lim_{n\to\infty}\left(1 + \frac{x}{n}\right)^n$$

show that

$$\lim_{n\to\infty}\left(1 + \frac{3}{4n}\right)^n = 4 e^{3/2}$$

I know im missing something stupid probably, just some clever little analysis trick should do the job.

  • 5
    $\begingroup$ I would rather have expected $e^{\frac 34}$ ... $\endgroup$ – Hagen von Eitzen Jan 5 at 23:40
  • $\begingroup$ This looks false. $(1+3/(4n))^n$ has the same limit at $\infty$ at $((1+1/n)^n)^{3/4}$ which converges to $e^{3/4}$. $\endgroup$ – Mindlack Jan 5 at 23:42
  • $\begingroup$ ok I went through this thought process too, so it is most likely a typo. These curriculum sheets are the only ones with no answer key as they have been custom built for a very advanced 12 year old student. Thanks so much, we spent so much time spinning in circles. $\endgroup$ – Hossien Sahebjame Jan 6 at 0:10

Can do let $k = \frac{4}{3}n $. Then as $n \to \infty$ certainly $k \to \infty$ and moreover

$$ \lim_{k \to \infty} (1 + 1/k)^{3/4 k } = (\lim_{k \to \infty} (1 + 1/k)^{1/k})^{3/4} = e^{3/4} $$


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