Let's explore the limit using an intuitive approach.

We are asked to find the limit:

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

Step 1: Substitute $x = 2$

First, let's try plugging $x = 2$ directly into the expression:

$\frac{2^2 - 4}{2 - 2} = \frac{4 - 4}{0} = \frac{0}{0}$

This results in the indeterminate form $\frac{0}{0}$. It means we cannot directly compute the limit by simple substitution, but it also suggests that we might be able to simplify the expression.

Step 2: Simplify the expression

The numerator, $x^2 - 4$, is a difference of squares. We can factor it as:

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

So, the original expression becomes:

$\frac{x^2 - 4}{x - 2} = \frac{(x - 2)(x + 2)}{x - 2}$

Now, for $x \neq 2$, we can cancel the common factor $x - 2$ from both the numerator and the denominator:

$\frac{(x - 2)(x + 2)}{x - 2} = x + 2 \quad \text{for} \quad x \neq 2$

Step 3: Evaluate the limit of the simplified expression

Now that we have simplified the expression to $x + 2$, we can substitute $x = 2$ without any issues:

$\lim_{x \to 2} (x + 2) = 2 + 2 = 4$

Conclusion:

Therefore, the limit exists and its value is:

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

Why does the limit exist?

The indeterminate form $\frac{0}{0}$ indicates that the function has a removable discontinuity at $x = 2$. After simplifying the expression, the discontinuity is "removed," and we are left with a continuous function $x + 2$, which has no problems at $x = 2$. This is why the limit exists and equals 4.

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Would you like further details or have any questions?

Here are 5 related questions:

1. How do we handle other indeterminate forms like $\frac{0}{0}$?
2. What is the difference between a removable and non-removable discontinuity?
3. How does factoring help in solving limits involving polynomials?
4. Can limits be used to define continuity of a function?
5. How does L'Hopital's rule apply to indeterminate forms like $\frac{0}{0}$?

Tip: When dealing with indeterminate forms, always look for algebraic simplifications such as factoring, rationalizing, or canceling common terms.