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Part 1

The federal tax rate for heads of household is given in the graph shown to the right. Use the graph to find

ModifyingBelow lim With x right arrow 8466 Superscript minus

​r(x),

ModifyingBelow lim With x right arrow 8466 Superscript plus

​r(x), 

and

ModifyingBelow lim With x right arrow 8466

​r(x).

The graph depicts a tax rate function \( r(x) \), where \( x \) is the taxable income for heads of household. We're tasked with finding the following limits related to the point \( x = 8466 \):

1. **Left-hand limit:** \( \lim_{x \to 8466^-} r(x) \)
2. **Right-hand limit:** \( \lim_{x \to 8466^+} r(x) \)
3. **Two-sided limit:** \( \lim_{x \to 8466} r(x) \)

### Analyzing the Graph:
- \( x = 8466 \) corresponds to a boundary where the tax rate seems to change.
- To find these limits, we observe the values of \( r(x) \) approaching \( x = 8466 \) from both directions.

#### 1. Left-hand limit \( \lim_{x \to 8466^-} r(x) \):
- For values of \( x \) just less than 8466, the tax rate on the graph appears to be 10%.
- Thus, \( \lim_{x \to 8466^-} r(x) = 10\% \).

#### 2. Right-hand limit \( \lim_{x \to 8466^+} r(x) \):
- For values of \( x \) just greater than 8466, the tax rate jumps to 12%.
- So, \( \lim_{x \to 8466^+} r(x) = 12\% \).

#### 3. Two-sided limit \( \lim_{x \to 8466} r(x) \):
- Since the left-hand and right-hand limits are not equal (10% and 12%, respectively), the two-sided limit **does not exist**.

### Summary of Limits:
- \( \lim_{x \to 8466^-} r(x) = 10\% \)
- \( \lim_{x \to 8466^+} r(x) = 12\% \)
- \( \lim_{x \to 8466} r(x) \) does not exist because the function is discontinuous at \( x = 8466 \).

Let me know if you'd like further details or have any questions.

Here are five related questions for further exploration:
1. How would the tax rate change as we approach another boundary like \( x = 30382 \)?
2. How can we mathematically define discontinuities in functions like these?
3. What is the significance of different types of limits in analyzing tax brackets?
4. How does the jump discontinuity reflect changes in real-life tax systems?
5. How would you express these limits in terms of a piecewise function?

**Tip:** When dealing with discontinuities in functions, always check the left-hand and right-hand limits to understand how the function behaves at specific points.

The federal tax rate for heads of household is given in the graph shown to the right. Use the graph to find lim(x → 8466-) r(x), lim(x → 8466+) r(x), and lim(x → 8466) r(x).

Explore how to calculate left-hand, right-hand, and two-sided limits for a tax rate function at specific income levels, including an analysis of discontinuities at x = 8466. This detailed guide covers key calculus concepts relevant to understanding federal tax brackets.

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