x 2 −4 x−2 ​

The given expression is:

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

Step 1: Simplify the expression

Notice that the numerator $x^2 - 4$ is a difference of squares:

$x^2 - 4 = (x - 2)(x + 2).$

Substitute this back into the expression:

$\lim_{x \to 2} \frac{(x - 2)(x + 2)}{x - 2}.$

Step 2: Cancel the common factor

For $x \neq 2$, the term $x - 2$ in the numerator and denominator cancels out:

$\lim_{x \to 2} (x + 2).$

Step 3: Evaluate the limit

Now, substitute $x = 2$ into the simplified expression:

$x + 2 = 2 + 2 = 4.$

Thus, the value of the limit is:

$\boxed{4}.$

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Would you like further clarification or additional examples?

Related Questions

1. How do you handle indeterminate forms like $\frac{0}{0}$ when evaluating limits?
2. Can L'Hôpital's Rule be used for this problem? Why or why not?
3. What are other examples of simplifying algebraic expressions for limits?
4. What is the difference between a one-sided and a two-sided limit?
5. How can this problem be extended to explore continuity?

Tip

Always look for factorizations or common terms when evaluating limits to simplify expressions!