https://file.aiquickdraw.com/mathbot/file/1726624044896pjacc5j7.jpg

Let's analyze the two limits based on the images provided in the homework assignment.

1. **Limit 1:**
   \[
   \lim_{x \to 1} \frac{x^2 + 3x + 2}{x + 1}
   \]
   From the graph, we observe that the function seems to be a straight line excluding the point at \(x = -1\) where there's a hole. This suggests that \(x + 1\) is a factor of the numerator \(x^2 + 3x + 2\). Factoring the numerator:
   \[
   x^2 + 3x + 2 = (x + 1)(x + 2)
   \]
   Therefore, the function simplifies to:
   \[
   \frac{x^2 + 3x + 2}{x + 1} = x + 2, \quad \text{for } x \neq -1
   \]
   Thus, at \(x = 1\), the function is well-defined and:
   \[
   \lim_{x \to 1} (x + 2) = 1 + 2 = 3
   \]

2. **Limit 2:**
   \[
   \lim_{x \to 0^-} f(x) \text{ and } \lim_{x \to 0^+} f(x) \text{ where } f(x) = \begin{cases} 
   3x + 1 & \text{if } x \leq 0 \\
   2 & \text{if } x > 0
   \end{cases}
   \]
   From the piecewise definition and the graph:
   - As \(x\) approaches 0 from the left (\(0^-\)), the function is defined by \(3x + 1\). Plugging in \(x = 0\):
     \[
     \lim_{x \to 0^-} (3x + 1) = 3 \cdot 0 + 1 = 1
     \]
   - As \(x\) approaches 0 from the right (\(0^+\)), the function value is constantly 2.
     \[
     \lim_{x \to 0^+} f(x) = 2
     \]
   Because the left-hand limit (\(1\)) does not equal the right-hand limit (\(2\)), the overall limit at \(x = 0\) does not exist.

These interpretations confirm the behavior of the functions as illustrated by the graphs in your homework.

Do you want to explore these concepts further or have any questions about this analysis?

**Related Questions:**
1. What would happen if the function \(f(x)\) were defined differently for \(x > 0\) in problem 2?
2. How do the definitions of one-sided limits contribute to understanding discontinuities in functions?
3. Can you find another example where a polynomial is simplified due to factorization, similar to problem 1?
4. What are other common types of discontinuities in functions beyond what's shown here?
5. How might the graph look if the numerator in problem 1 had a different factor that did not cancel with the denominator?

**Tip:** When analyzing limits graphically, look for points where the function isn't continuous, like holes or jumps, as these often signal where limits might not exist or need closer examination through algebraic simplification or piecewise definitions.

In Exercises 1 and 2, estimate and/or calculate the limits by using the graphs and/or by using the definition of the limit, or explain why the limit does not exist.

Explore how to estimate and calculate limits involving polynomial functions and piecewise functions. This problem dives into the fundamentals of algebraic manipulation and graphical understanding of limits, covering factoring polynomials and one-sided limits. Suitable for college-level students in Calculus 1.

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