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.