To prove that the functions $f(x) = 3x^2 + 3$, $g(x) = 2x^2 + 3$, and $h(x) = -2x^2 + 3$ can be used with the Squeeze Theorem, we need to verify if $g(x)$ is squeezed between $f(x)$ and $h(x)$ as $x \to a$ (where $a$ is some value, often zero or infinity in standard applications).

Step 1: Establish the inequalities

We need to check if the function $g(x)$ satisfies: $h(x) \leq g(x) \leq f(x)$

for all $x$ in some interval around the point of interest, except possibly at that point.

Substituting the functions:

$-2x^2 + 3 \leq 2x^2 + 3 \leq 3x^2 + 3$

Let's verify these inequalities separately:

1. Check if $-2x^2 + 3 \leq 2x^2 + 3$:

   Simplify the inequality:

   $-2x^2 + 3 \leq 2x^2 + 3$

   Subtract $3$ from both sides:

   $-2x^2 \leq 2x^2$

   Divide by $2$:

   $-x^2 \leq x^2$

   This inequality holds true for all real values of $x$.

2. Check if $2x^2 + 3 \leq 3x^2 + 3$:

   Simplify the inequality:

   $2x^2 + 3 \leq 3x^2 + 3$

   Subtract $3$ from both sides:

   $2x^2 \leq 3x^2$

   Simplify:

   $0 \leq x^2$

   This inequality also holds true for all real values of $x$.

Since both inequalities are satisfied for all $x$, we have:

$h(x) \leq g(x) \leq f(x)$

Step 2: Apply the Squeeze Theorem

To apply the Squeeze Theorem, we need to evaluate the limits of $f(x)$ and $h(x)$ as $x \to a$. Let’s take $a = 0$ for simplicity.

$\lim_{x \to 0} f(x) = \lim_{x \to 0} (3x^2 + 3) = 3$

$\lim_{x \to 0} h(x) = \lim_{x \to 0} (-2x^2 + 3) = 3$

Since $h(x) \leq g(x) \leq f(x)$ and both outer limits are 3, the Squeeze Theorem states:

$\lim_{x \to 0} g(x) = 3$

Thus, $g(x)$ is squeezed between $f(x)$ and $h(x)$, and the limit is valid according to the Squeeze Theorem.

Would you like more details or have any questions about this proof?

Related Questions:

1. How does the Squeeze Theorem apply when limits approach infinity?
2. What happens if $h(x)$ and $f(x)$ have different limits?
3. Can the Squeeze Theorem be used for sequences, and if so, how?
4. How would this proof change if we took limits as $x \to \infty$?
5. How can the Squeeze Theorem help in proving trigonometric limits?

Tip: The Squeeze Theorem is very useful for finding limits when you can bound a function between two simpler functions whose limits you know!