<s>[INST] What am I doing wrong finding the limit of $\left(\frac{3x^2-x+1}{2x^2+x+1}\right)^\left(\frac{x^3}{1-x}\right)$? $$\lim_{x\to\infty}\left(\frac{3x^2-x+1}{2x^2+x+1}\right)^\left(\frac{x^3}{1-x}\right)$$
Divide by $x^2$, get
$$\lim_{x\to\infty}(1,5)^\infty=\infty$$
The answer in the book is $0$.
I've also tried substituting $x$ for $\frac{1}{h}$ where $h$ tends to zero and using some form of $(1+\frac{1}{n})^n$, and using the exponent rule, but everything lands me at infinity.
Do I misunderstand something fundamentally?
 [/INST] Because the natural logarithm is injective, it turns out that: $\lim_{x \to \infty} \frac{3x²-x+1}{2x²+x+1}^{\frac{x^3}{1-x}}=(\lim_{x \to \infty} \frac{3x²-x+1}{2x²+x+1})^{(\lim_{x \to \infty} \frac{x^3}{1-x})}$. Thus the limit is $(\frac{3}{2})^{-\infty}=\frac{1}{1.5^{\infty}}=0$.
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