For the function *f*(*x*)=−4(*x*−2)2, evaluate the left and right limits using the table shown below. If any of the limits do not exist and do not tend to ±∞, enter ∅ as your answer.

*x*−4(*x*−2)2*x*−4(*x*−2)2

1.9

−400

2.1

−400

1.99

−40000

2.01

−40000

1.999

−4000000

2.001

−4000000

1.9999

−400000000

2.0001

−400000000

1.99999

−40000000000

2.00001

−40000000000

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).

For the function f(x)=-4(x−2)^2, evaluate the left and right limits using the table shown below. If any of the limits do not exist and do not tend to ±∞, enter ∅ as your answer.

Learn how to evaluate the left and right limits of the function f(x)=-4(x−2)^2 as x approaches 2. This example demonstrates how the function approaches negative infinity from both sides, providing a detailed step-by-step solution. Suitable for students in Grades 11-12.

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