The limit problem in the image is:

$\lim_{{x \to \infty}} \frac{5x^3 - 3x^2 - 7x}{7 - 5x - 10x^3}$

To evaluate this limit, we observe that both the numerator and the denominator are polynomials. The highest degree term in both the numerator and denominator is $x^3$. As $x$ approaches infinity, the terms with the highest degree dominate the behavior of the expression.

Step-by-step solution:

1. Identify the highest degree term in both the numerator and denominator. The highest degree term is $x^3$ in both cases:

   • In the numerator: $5x^3$
   • In the denominator: $-10x^3$

2. Divide every term in the numerator and denominator by $x^3$ to simplify the limit:

$\lim_{{x \to \infty}} \frac{\frac{5x^3}{x^3} - \frac{3x^2}{x^3} - \frac{7x}{x^3}}{\frac{7}{x^3} - \frac{5x}{x^3} - \frac{10x^3}{x^3}}$

This simplifies to:

$\lim_{{x \to \infty}} \frac{5 - \frac{3}{x} - \frac{7}{x^2}}{\frac{7}{x^3} - \frac{5}{x^2} - 10}$

3. As $x \to \infty$, the terms with $x$ in the denominator ($\frac{3}{x}, \frac{7}{x^2}, \frac{7}{x^3}, \frac{5}{x^2}$) approach zero. So the expression reduces to:

$\lim_{{x \to \infty}} \frac{5}{-10}$

4. The final result is:

$\frac{5}{-10} = -\frac{1}{2}$

So, the value of the limit is:

$\boxed{-\frac{1}{2}}$

Let me know if you would like any further details or explanations!

Related Questions:

1. What happens when both the numerator and denominator approach infinity?
2. How do you handle limits involving polynomials of different degrees?
3. What is L'Hopital's Rule, and when would it apply to limits like this?
4. What happens if the degrees of the numerator and denominator are the same?
5. How would the answer change if the highest degree term in the denominator was positive?

Tip:

When evaluating limits involving polynomials, always look for the highest degree terms, as they dominate the behavior of the function at infinity.