I have to solve limit of rational function, but it turns out I do mistake somewhere. Where I do wrong? Does my calculations correct?
$$\lim_{x\to \infty}\frac{x^3-2x-1}{x^5-2x-1}$$ Step 1: $$\lim_{x\to\infty}\frac{x^5\left(\frac{1}{x^2}-\frac{2}{x^4}-\frac{1}{x^5}\right)}{x^5\left(1-\frac{2}{x^4}-\frac{1}{x^5}\right)}$$ Step 2: $$\lim_{x\to\infty}\frac{x^5\left(\frac{1}{x^2}-\frac{2}{x^4}-\frac{1}{x^5}\right)}{x^5\left(1-\frac{2}{x^4}-\frac{1}{x^5}\right)} = \frac{0-0-0}{1-0-0}=\frac{0}{1}=0$$