The given problem is:

$\lim_{x \to \infty} \frac{2 - 3x + x^3}{5x^2 - 7x + 1}.$

Solution:

To find the limit as $x \to \infty$, we analyze the degrees of the numerator and denominator.

1. The numerator is $2 - 3x + x^3$. The highest degree term is $x^3$.
2. The denominator is $5x^2 - 7x + 1$. The highest degree term is $5x^2$.

Observing growth:

As $x \to \infty$, the $x^3$ term in the numerator grows much faster than the $x^2$ term in the denominator.

Dividing by the highest power of $x$ in the denominator:

Divide each term in both the numerator and denominator by $x^2$ (the highest degree term in the denominator):

$\lim_{x \to \infty} \frac{2 - 3x + x^3}{5x^2 - 7x + 1} = \lim_{x \to \infty} \frac{\frac{2}{x^2} - \frac{3}{x} + x}{5 - \frac{7}{x} + \frac{1}{x^2}}.$

Simplify:

1. As $x \to \infty$, $\frac{2}{x^2} \to 0$, $\frac{3}{x} \to 0$, and $\frac{1}{x^2} \to 0$.
2. This leaves: $\lim_{x \to \infty} \frac{x}{5}.$

Final result:

$\frac{x}{5} \to \infty.$

So, the limit does not exist as it approaches infinity.

Would you like me to explore related examples or provide further clarification? Here are 5 related questions for practice:

1. What is the limit of $\lim_{x \to \infty} \frac{x^2 + 2x}{5x^2 + 1}?$
2. How do you determine if a rational function's limit approaches zero?
3. Can the degrees of numerator and denominator determine the behavior of the limit?
4. How does dividing by the highest power of $x$ simplify such problems?
5. Why does $\lim_{x \to \infty} \frac{1}{x}$ always approach zero?

Tip: Always compare the growth rates of the highest degree terms in the numerator and denominator for rational functions.