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.