The given mathematical expression is asking for the limit of a rational function as $x$ approaches infinity:

$\lim_{x \to \infty} \frac{-7x^2 + 3x - 6}{-2x + 1}$

Steps to solve:

1. Identify the degrees of the polynomials:

   • The numerator is $-7x^2 + 3x - 6$, which is a quadratic polynomial (degree 2).
   • The denominator is $-2x + 1$, which is a linear polynomial (degree 1).

2. Since the degree of the numerator (2) is greater than the degree of the denominator (1), the behavior of the function as $x$ approaches infinity will be dominated by the highest degree terms in both the numerator and the denominator. Specifically, it simplifies to:

$\frac{-7x^2}{-2x}$

3. Simplify the expression:

$\frac{-7x^2}{-2x} = \frac{7x}{2}$

As $x \to \infty$, $\frac{7x}{2}$ approaches infinity.

Conclusion:

$\lim_{x \to \infty} \frac{-7x^2 + 3x - 6}{-2x + 1} = \infty$

This means the value of the limit is infinite.

Let me know if you'd like more details or further clarifications!

Related questions:

1. What happens if the degrees of the numerator and denominator are the same?
2. How would you handle limits if $x \to -\infty$?
3. What if both the numerator and denominator were linear polynomials?
4. Can you compute limits approaching a specific finite number?
5. How would the limit change if the highest degree terms in both the numerator and denominator had different signs?

Tip:

When dealing with rational functions, always focus on the terms with the highest degrees as $x \to \infty$, since they dominate the behavior.