prove if functions f(x)=3x^2+3; g(x)=2x^2+3; h(x)=-2x^2+3 apply squeeze theorem

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!

Explore how the Squeeze Theorem applies to the quadratic functions f(x)=3x^2+3, g(x)=2x^2+3, and h(x)=-2x^2+3. This proof involves limits and inequalities, making it ideal for students in Grades 11-12 or at the college level.

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