<s>[INST] Limits of power of fractions I'm having trouble solving these limits:
$$\lim_{x\to \infty} \left (\frac{3-x}{2x+5}\right )^{3x}\\\lim_{x\to \infty} \left (\frac{2x+1}{3x-4}\right )^{1-2x}\\\lim_{x\to \infty} \left (\frac{2-3x}{1-3x}\right )^{x+4}$$
Are they even defined?
 [/INST] Hint: Knowing that $$e^a =\lim_{x\to \infty}\left(1+\frac{a}{x}\right)^x$$ 
show by yourself that: 
$$\lim_{x\to \infty} \left (\frac{3-x}{2x+5}\right )^{3x}~~~~~~~~DNE$$ 
$$\lim_{x\to \infty} \left (\frac{2x+1}{3x-4}\right )^{1-2x}= \lim_{x\to \infty} \left (\frac{3}{2}-\frac{11}{4x+2}\right )^{2x-1}=\infty$$ 
$$\lim_{x\to \infty} \left (\frac{2-3x}{1-3x}\right )^{x+4}=\lim_{x\to \infty} \left (1-\frac{1}{3x-1}\right )^{3x-1 \frac{x+4}{3x-1}} =e^{-\frac{1}{3}}$$
use similar idea as here Proving that $\lim_{n\to\infty}\left ( \frac{2n-1}{2n+3} \right )^n=e^{-2}$ without using de l‘Hôspital
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