1. **State the problem:** Find the limit $$\lim_{x \to 2} \frac{x^2 + 4}{x - 2}$$. 2. **Understand the problem:** We want to find the value that the expression approaches as $x$ gets closer to 2. 3. **Check direct substitution:** Substitute $x=2$ into the expression: $$\frac{2^2 + 4}{2 - 2} = \frac{4 + 4}{0} = \frac{8}{0}$$ which is undefined (division by zero). 4. **Analyze the behavior:** Since direct substitution leads to division by zero, the limit may be infinite or does not exist. 5. **Consider the numerator and denominator separately:** - Numerator at $x=2$ is $8$ (positive). - Denominator approaches $0$ from the left or right depending on $x$. 6. **Check the left-hand limit ($x \to 2^-$):** - Denominator $x-2$ is slightly negative. - Numerator is positive. - So the fraction approaches $+\infty$ or $-\infty$? Since numerator positive and denominator negative, fraction approaches $-\infty$. 7. **Check the right-hand limit ($x \to 2^+$):** - Denominator $x-2$ is slightly positive. - Numerator positive. - Fraction approaches $+\infty$. 8. **Conclusion:** Left and right limits are not equal, so the limit does not exist. **Final answer:** $$\lim_{x \to 2} \frac{x^2 + 4}{x - 2} \text{ does not exist}$$ because the left and right limits differ.