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It seems like you're working on a problem involving limits. The function given is:

\[
f(x) = \frac{x - 4}{(x - 1)^2}
\]

You're asked to compute the values of \( f(x) \) as \( x \) approaches 1 from both sides and use the table to help determine:

\[
\lim_{{x \to 1}} f(x)
\]

Here's how you can approach this:

1. **Substitute the given values of \( x \) into \( f(x) \):**
   - For \( x = 1.1, 1.01, 1.001, 1.0001 \), and similarly for values approaching from the left side.

2. **Evaluate \( f(x) \) at those points:**
   - For each value of \( x \), calculate \( f(x) = \frac{x - 4}{(x - 1)^2} \).

Let me help fill in some values to guide you. I'll start with calculating the function at specific values of \( x \).Here are the computed values of \( f(x) = \frac{x - 4}{(x - 1)^2} \) as \( x \) approaches 1 from both sides:

- **For values approaching from the right (x > 1):**
  - \( f(1.1) \approx -290 \)
  - \( f(1.01) \approx -29,900 \)
  - \( f(1.001) \approx -2,999,000 \)
  - \( f(1.0001) \approx -299,990,000 \)

- **For values approaching from the left (x < 1):**
  - \( f(0.9) \approx -310 \)
  - \( f(0.99) \approx -30,100 \)
  - \( f(0.999) \approx -3,000,000 \)
  - \( f(0.9999) \approx -300,010,000 \)

As you can see, as \( x \) gets closer to 1 from both sides, \( f(x) \) becomes very negative and large in magnitude. This suggests that \( \lim_{{x \to 1}} f(x) = -\infty \).

Would you like more details on the steps, or do you have any other questions?

Here are 5 related questions to expand your understanding:
1. What is the difference between a limit approaching from the left vs. the right?
2. Why does the function behave the way it does near \( x = 1 \)?
3. What would happen if we approached \( x = 4 \) instead of \( x = 1 \)?
4. Can you graph this function to visualize the behavior near \( x = 1 \)?
5. How does the denominator influence the rapid increase in magnitude near \( x = 1 \)?

**Tip:** When the denominator of a fraction approaches zero, the value of the function tends to grow infinitely, leading to vertical asymptotes in graphs.

Compute the values of f(x) = (x - 4) / (x - 1)^2 in the table to determine lim f(x) as x approaches 1.

This problem involves calculating the limit of the rational function f(x) = (x - 4) / (x - 1)^2 as x approaches 1. Students will use a table of values to observe the behavior of the function near x = 1 and determine if the limit exists. The solution explores the concept of limits and asymptotic behavior in calculus.

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