The problem you've provided is a limit calculation:

$\lim_{{x \to 2}} \frac{{x^2 - 4}}{{x - 2}}$

Solution

1. Factor the Numerator: The expression $x^2 - 4$ is a difference of squares, so it can be factored as: $x^2 - 4 = (x + 2)(x - 2)$

2. Rewrite the Limit: Substitute this factorization into the limit: $\lim_{{x \to 2}} \frac{{(x + 2)(x - 2)}}{{x - 2}}$

3. Cancel Common Terms: The $(x - 2)$ term in the numerator and denominator cancels out, simplifying the expression to: $\lim_{{x \to 2}} (x + 2)$

4. Evaluate the Limit: Now, substitute $x = 2$ into the simplified expression: $x + 2 = 2 + 2 = 4$

Final Answer

$\lim_{{x \to 2}} \frac{{x^2 - 4}}{{x - 2}} = 4$

Would you like a deeper explanation, or do you have any questions about this solution?

Here are some additional questions to extend your understanding:

1. How does factoring help in solving limits involving rational functions?
2. What is the difference of squares, and why is it useful in limit problems?
3. Why can't we directly substitute $x = 2$ at the start in this problem?
4. How would the solution change if the limit approached a different value, say $x \to 3$?
5. What other types of factorization techniques are helpful in solving limit problems?

Tip: When dealing with limits that result in an indeterminate form like $\frac{0}{0}$, look for opportunities to factor and simplify before evaluating the limit.