https://file.aiquickdraw.com/mathbot/file/1726665840169ub2qlzqj.jpg

From the image, the problem asks to find the following limits and function values using the provided graph for each given value of \( a \).

We are asked to evaluate the limits as \( x \to a \) from both the left and right, as well as the function value at \( x = a \), for two cases: \( a = 0 \) and \( a = 2 \).

Let’s focus on part **a.i** for \( a = 0 \):

The question asks for:
\[
\lim_{x \to 0^-} f(x)
\]
This represents the left-hand limit as \( x \) approaches 0 from the left side (negative direction).

### Examining the graph at \( a = 0 \):
- For \( x \to 0^- \) (left of 0), the function values are approaching **6** based on the direction of the arrow from the left side.

Thus, the limit is:
\[
\lim_{x \to 0^-} f(x) = 6
\]

Would you like to proceed with solving the other limits or ask more questions? 

Here are 5 related questions:
1. What is the right-hand limit \( \lim_{x \to 0^+} f(x) \)?
2. Does \( \lim_{x \to 0} f(x) \) exist, and why?
3. What is \( f(0) \), the actual function value at \( x = 0 \)?
4. What happens when \( a = 2 \) for the same limits and function value?
5. How can you determine when a limit does not exist from a graph?

**Tip:** When evaluating limits, always check for continuity by comparing the left-hand and right-hand limits. If they differ, the limit at that point does not exist.

Use the graph to find the following limits and function value for each value of a: i) lim f(x) as x→a⁻, ii) lim f(x) as x→a⁺, iii) lim f(x) as x→a, iv) f(a), for a = 0 and a = 2.

Learn how to find limits and function values using a graph. This problem focuses on left-hand and right-hand limits, continuity, and function values for a = 0 and a = 2. Suitable for high school students learning calculus.

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