The expression in the image is:

$\lim_{{x \to \sqrt[3]{2}}} \left( \frac{x^2}{2} - \frac{1}{x} \right)$

We need to evaluate this limit as $x$ approaches $\sqrt[3]{2}$.

Step-by-step solution:

1. Substitute $x = \sqrt[3]{2}$ directly into the expression:

   $\lim_{{x \to \sqrt[3]{2}}} \left( \frac{x^2}{2} - \frac{1}{x} \right)$

   Substituting $x = \sqrt[3]{2}$:

   • $\left(\sqrt[3]{2}\right)^2 = \sqrt[3]{4}$
   • Substituting this, we get:

   [ \frac{\sqrt[3]{4}}{2} - \frac{1}{\sqrt[3]{2}} ]

2. Simplify the expression:

   The final expression is:

   [ \frac{\sqrt[3]{4}}{2} - \frac{1}{\sqrt[3]{2}} ]

This is the simplified form of the limit, which can be approximated further if needed. Would you like to proceed with numerical approximation or leave it in this form?

Feel free to ask questions or request additional details. Here are some related questions to explore:

1. What techniques can be used to handle limits when direct substitution fails?
2. How do you simplify cube roots algebraically?
3. What are some common limit forms that require special techniques to evaluate?
4. How does L'Hopital's rule help in evaluating limits involving indeterminate forms?
5. Can we approximate the value of cube roots without a calculator?

Tip: When evaluating limits, always check if direct substitution works first. If it leads to an indeterminate form, consider using techniques like factorization or L'Hopital's rule.