https://file.aiquickdraw.com/mathbot/file/1727117412697qgq6llqw.jpg

Let's work through parts (e) and (f) of the image, where the answers were marked incorrect.

### (e) \(\lim_{x = -1} \left(3x^2 f(x)\right)\)

To solve this limit, let's break it down:

1. **Evaluate \( f(x) \) at \( x = -1 \):**
   From the graph of \( f(x) \), when \( x = -1 \), the value of \( f(x) \) is \( f(-1) = -1 \).

2. **Substitute into \( 3x^2 f(x) \):**
   Now, substitute \( x = -1 \) into \( 3x^2 f(x) \):
   \[
   3(-1)^2 f(-1) = 3(1)(-1) = -3.
   \]
Thus, the correct answer for part (e) is \( \boxed{-3} \), not \( -1 \).

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### (f) \(\lim_{x = -1} \sqrt{3 + f(x)}\)

Let's evaluate this limit step-by-step:

1. **Evaluate \( f(x) \) at \( x = -1 \):**
   Again, from the graph of \( f(x) \), when \( x = -1 \), the value of \( f(x) \) is \( f(-1) = -1 \).

2. **Substitute into \( \sqrt{3 + f(x)} \):**
   Now, substitute \( f(-1) = -1 \) into the expression \( \sqrt{3 + f(x)} \):
   \[
   \sqrt{3 + (-1)} = \sqrt{2}.
   \]
   The approximate value of \( \sqrt{2} \) is around \( 1.414 \), which is not 4.

Thus, the correct answer for part (f) is \( \boxed{\sqrt{2}} \), not 4.

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### Do you need further clarifications?

Here are 5 follow-up questions to enhance your understanding:

1. Why do limits fail to exist (as in part (b) and (d))?
2. How do you handle limits when dealing with piecewise functions like \( f(x) \) and \( g(x) \)?
3. How does the product of functions affect limits (as in part (c))?
4. What is the significance of using the square root in limits (as in part (f))?
5. How can you interpret limits graphically versus algebraically?

**Tip:** Always check the function values carefully on graphs when dealing with limits!

Evaluate the limits using the graphs of f(x) and g(x).

This problem focuses on evaluating limits using the graphs of f(x) and g(x). It involves applying the rules of limits, such as product rules and square root evaluations. Suitable for students in Grades 11-12 or college-level calculus.

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