1. **State the problem:** Find the limit $$\lim_{x\to 2}\frac{x^2-4}{x^2+9x+14}$$. 2. **Recall the formula and rules:** To find limits involving rational functions, first try direct substitution. If it results in an indeterminate form like $$\frac{0}{0}$$, factor and simplify. 3. **Direct substitution:** Substitute $$x=2$$: $$\frac{2^2-4}{2^2+9\cdot 2 +14} = \frac{4-4}{4+18+14} = \frac{0}{36} = 0$$. Since the denominator is not zero and numerator is zero, the limit is 0. 4. **Verification by factoring:** Factor numerator: $$x^2-4 = (x-2)(x+2)$$. Factor denominator: $$x^2+9x+14 = (x+7)(x+2)$$. 5. **Simplify the expression:** $$\frac{(x-2)(x+2)}{(x+7)(x+2)} = \frac{(x-2)\cancel{(x+2)}}{(x+7)\cancel{(x+2)}}$$ 6. **Simplified limit expression:** $$\lim_{x\to 2} \frac{x-2}{x+7}$$ 7. **Evaluate the simplified limit:** $$\frac{2-2}{2+7} = \frac{0}{9} = 0$$ **Final answer:** $$\boxed{0}$$