I see that you've uploaded a screenshot from your calculus assignment on limits. The focus seems to be on left-hand and right-hand limits, with an example involving a piecewise function.

In this assignment:

1. Left-hand limit as $x$ approaches a value $a$ from the left ($x \to a^-$) is denoted as: $\lim_{x \to a^-} f(x) = L$

2. Right-hand limit as $x$ approaches $a$ from the right ($x \to a^+$) is denoted as: $\lim_{x \to a^+} f(x) = L$ For the limit to exist at $x = a$, the left-hand and right-hand limits must be equal.

3. In the piecewise function in the graph:

   \frac{x^2 + 1}{x} & \text{if } x \leq 0, \\ x^2 & \text{if } x > 0. \end{cases}$$

The task asks you to explore:

• The left-hand and right-hand limits at $x = 1$ and whether they are equal, confirming if the limit exists.
• The same for $x = 0$.

Would you like further details on solving the limit or the specific values for this example?

Here are some guiding questions to help with this topic:

1. What conditions must be met for a two-sided limit to exist?
2. How do we calculate the limit of a piecewise function at a point where the function changes definition?
3. What happens if the left-hand and right-hand limits are different at a given point?
4. How can graphical behavior support or contradict the calculated limits?
5. What are some common pitfalls when working with limits involving piecewise functions?

Tip: Always check for continuity at the point where the function changes definition by comparing left-hand and right-hand limits.