The limit of the given expression as $x \to 2$ is $\frac{2}{9}$.

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Here are 5 related questions you might find useful:

1. How do we apply L'Hopital's Rule for limits involving indeterminate forms?
2. Can trigonometric functions like sine affect the limit outcome differently in certain cases?
3. How do we handle limits when the denominator is raised to a power?
4. What role does simplification play in finding limits of complex expressions?
5. How do we find the limit when square roots are involved in the numerator?

Tip: Always check for indeterminate forms (like $\frac{0}{0}$) when evaluating limits, as these might indicate the need for L'Hopital's Rule or algebraic manipulation.