1. **State the problem:** Determine if the function $$f(x) = \frac{x^2 - x - 2}{x - 2}$$ is continuous at $$x=2$$. 2. **Recall the definition of continuity at a point:** A function $$f$$ is continuous at $$x=a$$ if: - $$f(a)$$ is defined. - The limit $$\lim_{x \to a} f(x)$$ exists. - $$\lim_{x \to a} f(x) = f(a)$$. 3. **Check if $$f(2)$$ is defined:** The denominator $$x-2$$ becomes zero at $$x=2$$, so $$f(2)$$ is undefined. 4. **Simplify the function where possible:** Factor the numerator: $$x^2 - x - 2 = (x - 2)(x + 1)$$. So, $$f(x) = \frac{(x - 2)(x + 1)}{x - 2}$$ for $$x \neq 2$$. Canceling $$x-2$$ (except at $$x=2$$): $$f(x) = x + 1$$ for $$x \neq 2$$. 5. **Find the limit as $$x \to 2$$:** $$\lim_{x \to 2} f(x) = \lim_{x \to 2} (x + 1) = 2 + 1 = 3$$. 6. **Conclusion:** Since $$f(2)$$ is not defined, the function is not continuous at $$x=2$$. **Final answer:** $$f(x)$$ is discontinuous at $$x=2$$ because it is not defined there.