Let g, of, xg(x) and h, of, xh(x) be defined by g, of, x, equals, minus, 2, x, squared, plus, 4, xg(x)=−2x 
2
 +4x and h, of, x, equals, 16, x, plus, 18h(x)=16x+18. If f, of, xf(x) is a function such that g, of, x, is less than or equal to, f, of, x, is less than or equal to, h, of, xg(x)≤f(x)≤h(x), find limit, start subscript, x, right arrow, minus, 3, end subscript, f, of, x 
x→−3
lim
​	
 f(x), or state that it cannot be determined.

We are given two functions, \( g(x) = -2x^2 + 4x \) and \( h(x) = 16x + 18 \), along with an inequality \( g(x) \leq f(x) \leq h(x) \). We are tasked with finding:

\[
\lim_{x \to -3} f(x)
\]

### Step 1: Compute \( \lim_{x \to -3} g(x) \)
We begin by finding the limit of \( g(x) \) as \( x \) approaches -3:

\[
g(x) = -2x^2 + 4x
\]

Substitute \( x = -3 \) into \( g(x) \):

\[
g(-3) = -2(-3)^2 + 4(-3) = -2(9) - 12 = -18 - 12 = -30
\]

Thus, \( \lim_{x \to -3} g(x) = -30 \).

### Step 2: Compute \( \lim_{x \to -3} h(x) \)
Next, we find the limit of \( h(x) \) as \( x \) approaches -3:

\[
h(x) = 16x + 18
\]

Substitute \( x = -3 \) into \( h(x) \):

\[
h(-3) = 16(-3) + 18 = -48 + 18 = -30
\]

Thus, \( \lim_{x \to -3} h(x) = -30 \).

### Step 3: Apply the Squeeze Theorem
Since \( g(x) \leq f(x) \leq h(x) \), and both \( \lim_{x \to -3} g(x) = -30 \) and \( \lim_{x \to -3} h(x) = -30 \), by the **Squeeze Theorem**, we can conclude that:

\[
\lim_{x \to -3} f(x) = -30
\]

Thus, the limit is \( -30 \).

### Conclusion:
\[
\lim_{x \to -3} f(x) = -30
\]

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Would you like more details or have any questions?

Here are five related questions to explore:
1. What is the Squeeze Theorem, and how is it used in limit problems?
2. How can we find the limit of a quadratic function?
3. What happens when a function is bounded by two other functions?
4. How can we verify if the Squeeze Theorem applies in other scenarios?
5. How do you compute limits for piecewise functions?

**Tip:** When dealing with limits involving inequalities, always check if the Squeeze Theorem can help simplify the problem.

Let g(x) and h(x) be defined by g(x) = −2x^2 + 4x and h(x) = 16x + 18. If f(x) is a function such that g(x) ≤ f(x) ≤ h(x), find lim(x→−3) f(x), or state that it cannot be determined.

Find the limit of f(x) as x approaches -3 using the Squeeze Theorem. This problem involves quadratic and linear functions with limits and is suitable for students in Grades 10-12. Explore the step-by-step solution, applying the Squeeze Theorem.

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