lim_{x→2} (x²−4)/(x−2) =

To find the limit $$\\lim_{x \to 2} \\frac{x^2-4}{x-2}$$, follow these steps: 1. **Check for Direct Substitution:** Substitute $x = 2$ into the expression: $$\\frac{2^2 - 4}{2 - 2} = \\frac{4 - 4}{0} = \\frac{0}{0}$$ This is an **indeterminate form**, which means we must simplify the expression to find the limit. 2. **Factor the Numerator:** The numerator $x^2 - 4$ is a difference of squares ($a^2 - b^2 = (a-b)(a+b)$): $$x^2 - 4 = (x - 2)(x + 2)$$ 3. **Simplify the Fraction:** Substitute the factored form back into the limit: $$\\lim_{x \to 2} \\frac{(x - 2)(x + 2)}{x - 2}$$ As long as $x \\neq 2$, we can cancel the common term $(x - 2)$: $$\\lim_{x \to 2} (x + 2)$$ 4. **Evaluate the Limit:** Now, substitute $x = 2$ into the simplified expression: $$2 + 2 = 4$$ Therefore, the limit is **4**.