Limit Function 5287F5

1. **State the problem:** We want to find the limit of the function $$f(x)$$ as $$x$$ approaches $$3$$. 2. **Understand the function:** The function is defined as: $$ f(x) = \begin{cases} x^2 + 1 & \text{if } x < 3 \\ 5 & \text{if } x = 3 \end{cases} $$ 3. **Recall the limit definition:** The limit $$\lim_{x \to 3} f(x)$$ is the value that $$f(x)$$ approaches as $$x$$ gets arbitrarily close to $$3$$ from both sides. 4. **Calculate the left-hand limit:** Since for $$x < 3$$, $$f(x) = x^2 + 1$$, we compute: $$ \lim_{x \to 3^-} f(x) = \lim_{x \to 3^-} (x^2 + 1) = 3^2 + 1 = 9 + 1 = 10 $$ 5. **Calculate the right-hand limit:** The function is not explicitly defined for $$x > 3$$, so we consider the right-hand limit undefined or not given. 6. **Check the value at $$x=3$$:** $$f(3) = 5$$ 7. **Conclusion:** Since the left-hand limit $$10$$ does not equal the function value at $$3$$ which is $$5$$, and the right-hand limit is not defined, the limit $$\lim_{x \to 3} f(x)$$ does not exist. **Final answer:** $$\lim_{x \to 3} f(x) \text{ does not exist}$$