From the image, the mathematical problem is to evaluate the limit:

$\lim_{x \to \infty} \frac{4x + 6\sqrt{x^6 - 7x + 1}}{2x - 1}$

Let me solve it step by step:

Step 1: Factor the dominant terms

As $x \to \infty$, the highest power of $x$ dominates in both the numerator and denominator.

Numerator:

The dominant term in the numerator is $6\sqrt{x^6} = 6x^3$ because $\sqrt{x^6 - 7x + 1} \approx \sqrt{x^6}$.

Thus: $4x + 6\sqrt{x^6 - 7x + 1} \approx 6x^3 + 4x.$

Denominator:

The dominant term in the denominator is $2x$.

Thus: $2x - 1 \approx 2x.$

Step 2: Simplify the fraction

Divide both numerator and denominator by $x^3$, the highest power of $x$ in the expression.

$\frac{4x + 6\sqrt{x^6 - 7x + 1}}{2x - 1} \approx \frac{\frac{4x}{x^3} + \frac{6\sqrt{x^6}}{x^3}}{\frac{2x}{x^3} - \frac{1}{x^3}}.$

Simplify: $\frac{\frac{4}{x^2} + 6}{\frac{2}{x^2}}.$

Step 3: Evaluate the limit as $x \to \infty$

As $x \to \infty$, terms involving $\frac{1}{x^2}$ vanish because they approach 0.

Thus: $\lim_{x \to \infty} \frac{\frac{4}{x^2} + 6}{\frac{2}{x^2}} = \frac{6}{0}.$