<s>[INST] Evaluating the limit $\lim_{x\to\infty}\left(\frac {x^5+\pi x^4+e}{x^5+ex^4+\pi}\right)^x$ How can I evaluate the following limit?
$$\lim_{x\to\infty}\left(\frac {x^5+\pi x^4+e}{x^5+ex^4+\pi}\right)^x$$
To solve this limit, I divided the numerator and denominator by $x^5$ and I got
$$\lim_{x\to\infty}\left(\frac {1+\frac {1}{x}\pi+\frac {1}{x^5}e}{1+\frac {1}{x}e+\frac {1}{x^5}\pi}\right)^x$$
But I realized when $x\to\infty$  the expression is still indeterminate.
I tried changing the $x=\frac {1}{y}$ variable to be able to apply Lophital's rule, but I couldn't get a useful result.
 [/INST] $$\lim_{x\to\infty}\left(\frac {1+\frac {1}{x}\pi+\frac {1}{x^5}e}{1+\frac {1}{x}e+\frac {1}{x^5}\pi}\right)^x$$
Arrived at this point, you can use geometric series:
$$\left(\frac {1+\frac {1}{x}\pi+\frac {1}{x^5}e}{1+\frac {1}{x}e+\frac {1}{x^5}\pi}\right)^x\sim\left(\left(1+\frac {1}{x}\pi+\frac {1}{x^5}e\right)\left(1-\frac {1}{x}e-\frac {1}{x^5}\pi \right)\right)^x\sim \left(1 +\frac{\pi-e}{x}\right)^x \to e^{\pi-e}$$
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