1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{\ln\left(x - \sqrt[3]{2x} - 3\right)}{\sin\left(\frac{\pi x}{2}\right) - \sin\left(\pi(x - 1)\right)}$$ 2. **Analyze the expression:** - The numerator is a logarithm of an expression involving $x$ and a cube root. - The denominator is a difference of sine functions. 3. **Check the form by direct substitution:** - Substitute $x=2$: - Inside the logarithm: $2 - \sqrt[3]{4} - 3 = 2 - \sqrt[3]{4} - 3 = -1 - \sqrt[3]{4}$ which is negative, so the logarithm is undefined at $x=2$. 4. **Rewrite the numerator inside the logarithm:** Let $$f(x) = x - \sqrt[3]{2x} - 3$$ We want to find the behavior of $f(x)$ near $x=2$. 5. **Evaluate $f(2)$:** $$f(2) = 2 - \sqrt[3]{4} - 3 = -1 - \sqrt[3]{4} < 0$$ Since the argument of the logarithm is negative near $x=2$, the limit does not exist in the real numbers. 6. **Conclusion:** The logarithm is not defined for values near $x=2$ because the argument is negative. Therefore, the limit does not exist in the real domain. **Final answer:** The limit does not exist (DNE) in real numbers.