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give me the Lim x -> 3 from the left side of f(g(x)). After give me the lim x -> 0 from the left side of f(g(x)). Give me your reasoning

Let's evaluate the two limits step by step based on the given graphs.

### 1. \(\lim_{x \to 3^-} f(g(x))\)
- **Step 1**: First, find \( \lim_{x \to 3^-} g(x) \).
  From the graph of \(g(x)\) (the blue graph), as \(x\) approaches 3 from the left, \(g(x)\) approaches 0. We can see that the value of \(g(x)\) approaches 0 from above as \(x\) approaches 3 from the left.

- **Step 2**: Next, evaluate \( f(g(x)) \) using \( g(x) \to 0 \).
  Now, look at the graph of \(f(x)\) (the red graph). At \(x = 0\), there is a hole in the graph, but the value of \(f(x)\) approaches 1 as \(x\) approaches 0. Thus, we have:
  \[
  \lim_{x \to 3^-} f(g(x)) = f(0) = 1
  \]

### 2. \(\lim_{x \to 0^-} f(g(x))\)
- **Step 1**: First, find \( \lim_{x \to 0^-} g(x) \).
  From the graph of \(g(x)\), as \(x\) approaches 0 from the left, \(g(x)\) approaches \(0\) as well. The function seems to behave continuously at this point.

- **Step 2**: Evaluate \(f(g(x))\) using \(g(x) \to 0\).
  Again, for \(f(x)\), as \(x \to 0\), \(f(x)\) approaches 1 (even though there is a hole at \(f(0)\), the value approaches 1 from both sides).

Thus, we get:
\[
\lim_{x \to 0^-} f(g(x)) = 1
\]

### Summary:
- \(\lim_{x \to 3^-} f(g(x)) = 1\)
- \(\lim_{x \to 0^-} f(g(x)) = 1\)

Would you like more details on any of these steps or further explanations?

Here are 5 relative questions:
1. What is the limit of \(g(x)\) as \(x \to 3^+\)?
2. How would you evaluate \(f(g(x))\) for \(x \to 1^-\)?
3. What happens if the limit does not exist for \(f(x)\) at certain points?
4. How does continuity of a function impact its limit at a point?
5. How would the result change if there was no hole at \(f(0)\)?

**Tip**: Always check the behavior of both the outer and inner functions when dealing with composite limits.

This problem covers how to evaluate the left-hand limits of composite functions using f(g(x)) as x approaches 3 and 0. The solution involves analyzing both the inner and outer functions graphically. Suitable for Calculus I students.

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