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The limit provided is:

\[
\lim_{{x \to 2}} \frac{{2^x - x^2}}{{x - 2}}
\]

### Step-by-Step Solution:

1. **Substitution Check**: 
   Let's first substitute \(x = 2\) directly into the expression to see if we get a determinate form:
   
   \[
   \frac{{2^2 - 2^2}}{{2 - 2}} = \frac{{4 - 4}}{{0}} = \frac{0}{0}
   \]

   This gives us an indeterminate form \(\frac{0}{0}\), so we need to apply another method to evaluate the limit.

2. **L'Hopital's Rule**: 
   Since we have a \(\frac{0}{0}\) form, we can apply L'Hopital's Rule. This rule states that for limits of the form \(\frac{0}{0}\) or \(\frac{\infty}{\infty}\), we can differentiate the numerator and denominator and then take the limit of the new expression.

   Let's differentiate the numerator and the denominator with respect to \(x\).

   - Numerator: \(2^x - x^2\)
     - The derivative of \(2^x\) is \(2^x \ln 2\).
     - The derivative of \(-x^2\) is \(-2x\).
     So, the derivative of the numerator is:
     \[
     \frac{d}{dx}(2^x - x^2) = 2^x \ln 2 - 2x
     \]

   - Denominator: \(x - 2\)
     - The derivative of \(x - 2\) is 1.

3. **New Limit**:
   After applying L'Hopital's Rule, the limit becomes:
   \[
   \lim_{{x \to 2}} \frac{2^x \ln 2 - 2x}{1}
   \]
   Now, substitute \(x = 2\) into this expression:
   \[
   2^2 \ln 2 - 2(2) = 4 \ln 2 - 4
   \]

4. **Final Answer**:
   The value of the limit is:
   \[
   4(\ln 2 - 1)
   \]

### Conclusion:
\[
\lim_{{x \to 2}} \frac{{2^x - x^2}}{{x - 2}} = 4(\ln 2 - 1)
\]

Would you like further details or explanations on any step?

---

Here are some related questions for deeper understanding:

1. How does L'Hopital's Rule work in other indeterminate forms like \(\frac{\infty}{\infty}\)?
2. What are some alternatives to L'Hopital's Rule for evaluating limits?
3. Can you explain why \(2^x\) differentiates to \(2^x \ln 2\)?
4. What is the geometric interpretation of limits in calculus?
5. How do exponential and polynomial functions behave near points of indeterminacy?

**Tip**: Always check for an indeterminate form before applying L'Hopital's Rule.

Learn how to evaluate the limit lim(x → 2) (2^x - x^2) / (x - 2) using L'Hopital's Rule. This problem covers key concepts in limits and indeterminate forms and applies L'Hopital's Rule for a solution. Suitable for advanced high school students studying calculus.

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