We are asked to evaluate the left and right limits of the function $f(x) = -4(x - 2)^2$ as $x$ approaches 2. The table you've provided shows values of $f(x)$ near $x = 2$, and we can use those to estimate the left-hand and right-hand limits.

Left-hand limit:

The left-hand limit refers to the behavior of the function as $x$ approaches 2 from values less than 2. This corresponds to values like 1.9, 1.99, 1.999, etc.

From the table:

• As $x = 1.9$, $f(x) = -400$
• As $x = 1.99$, $f(x) = -40,000$
• As $x = 1.999$, $f(x) = -4,000,000$
• As $x = 1.9999$, $f(x) = -400,000,000$
• As $x = 1.99999$, $f(x) = -40,000,000,000$

We see that as $x$ approaches 2 from the left, the values of $f(x)$ grow increasingly negative, becoming very large in magnitude (approaching $-\infty$).

Thus, the left-hand limit is:

$\lim_{x \to 2^-} f(x) = -\infty$

Right-hand limit:

The right-hand limit refers to the behavior of the function as $x$ approaches 2 from values greater than 2. This corresponds to values like 2.1, 2.01, 2.001, etc.

From the table:

• As $x = 2.1$, $f(x) = -400$
• As $x = 2.01$, $f(x) = -40,000$
• As $x = 2.001$, $f(x) = -4,000,000$
• As $x = 2.0001$, $f(x) = -400,000,000$
• As $x = 2.00001$, $f(x) = -40,000,000,000$

We see that as $x$ approaches 2 from the right, the values of $f(x)$ also become increasingly negative, growing in magnitude (approaching $-\infty$).

Thus, the right-hand limit is:

$\lim_{x \to 2^+} f(x) = -\infty$

Conclusion:

Both the left-hand and right-hand limits approach $-\infty$ as $x \to 2$.

Final Answer:

• Left-hand limit: $-\infty$
• Right-hand limit: $-\infty$

Would you like further explanation or have any questions on this? Here are some related questions you might find interesting:

1. What happens to the function as $x$ gets closer to 2 from both sides in general?
2. How would the limits behave if the coefficient of $(x - 2)^2$ was positive?
3. What does it mean when limits approach $-\infty$ or $+\infty$?
4. How can you graph the function $f(x) = -4(x - 2)^2$ to visualize these limits?
5. Can you find the general form of the limit of a quadratic function like this as $x$ approaches a specific value?

Tip: The behavior of quadratic functions around specific points often results in limits approaching $\infty$ or $-\infty$, particularly when the coefficient is negative (indicating a downward-facing parabola).