How to evaluate a limit similar to Euler identity?

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I have a limit as $x$ approaches infinity of $(1+4/(7x))^x$. Solutions I saw included transforming it into $e$ to the power of natural $\log$ etc. but I've seen a way simpler transformation but can't remember it. Any help appreciated.

Answer 1

If you're familiar with L'Hospital:

$$\lim_{x\to \infty}\ln((1+4/7x)^x)=\lim_{x\to \infty}x\ln(1+4/7x)=\lim_{x\to \infty}\frac{\ln(1+4/7x)}{1/x}=\lim_{x\to \infty}\frac{-4/x(7x+4)}{-1/x^2}=\lim_{x\to \infty}4x/(7x+4)=4/7 $$ This implies: $$\lim_{x\to \infty}(1+4/(7x))^x=e^{\lim_{x\to \infty}\ln((1+4/7x)^x)}=e^{4/7}$$

This doesn't use the property that $\lim_{x\to \infty}(1+t/x)^x=e^t$.