The image asks to solve the following limit problem:

$\lim_{x \to \infty} \left( \frac{1}{2} - x \cdot \tan\left(\frac{8}{x}\right) \right)$

Let's break it down and solve step by step.

Step 1: Approximate $\tan\left(\frac{8}{x}\right)$ as $x \to \infty$

For very large values of $x$, $\frac{8}{x}$ approaches zero. We can use the approximation for small angles: $\tan\left(\frac{8}{x}\right) \approx \frac{8}{x}$ This approximation holds because $\tan(\theta) \approx \theta$ for $\theta$ close to zero.

Step 2: Substitute the approximation into the limit expression

Now substitute $\tan\left(\frac{8}{x}\right) \approx \frac{8}{x}$ into the limit expression: $\lim_{x \to \infty} \left( \frac{1}{2} - x \cdot \frac{8}{x} \right)$ Simplifying the expression: