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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:
   \[
   f(x) = \begin{cases}
   \frac{x^2 + 1}{x} & \text{if } x \leq 0, \\
   x^2 & \text{if } x > 0.
   \end{cases}
   \]

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.

Left-hand and right-hand limits for the given piecewise function. Determine whether the limit exists at x = 1 and x = 0.

Explore the concept of left-hand and right-hand limits in calculus, using a piecewise function example. Learn to determine if the limits exist at x = 1 and x = 0. This problem is suitable for undergraduate calculus students.

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