Evaluate the following limits:
`lim_(xrarr2)((3^(x)+3^(3-x)-12)/(3^(3-x)-3^(x/2)))`

Explanation

To evaluate the limit

\[ \lim_{x \to 2} \frac{3^x + 3^{3-x} - 12}{3^{3-x} - 3^{x/2}}, \]

we will follow these steps:

### Step 1: Substitute \( x = 2 \)

First, we substitute \( x = 2 \) into the expression to check if we get a determinate form.

\[ 3^2 + 3^{3-2} - 12 = 9 + 3 - 12 = 0, \]
\[ 3^{3-2} - 3^{2/2} = 3 - 3 = 0. \]

Since both the numerator and denominator evaluate to 0, we have an indeterminate form \( \frac{0}{0} \), so we can apply L'Hôpital's Rule or simplify further.

### Step 2: Rewrite the expression

We can rewrite the limit as:

\[ \lim_{x \to 2} \frac{3^x + 3^{3-x} - 12}{3^{3-x} - 3^{x/2}}. \]

### Step 3: Change of variable

Let us set \( t = 3^{x/2} \). Then, as \( x \to 2 \), \( t \to 3 \).

We can express \( 3^x \) and \( 3^{3-x} \) in terms of \( t \):

\[ 3^x = t^2, \quad 3^{3-x} = \frac{27}{t^2}. \]

Substituting these into the limit gives:

\[ \lim_{t \to 3} \frac{t^2 + \frac{27}{t^2} - 12}{\frac{27}{t^2} - t}. \]

### Step 4: Simplify the expression

Now we simplify the numerator and denominator:

**Numerator:**
\[ t^2 + \frac{27}{t^2} - 12 = \frac{t^4 - 12t^2 + 27}{t^2}. \]

**Denominator:**
\[ \frac{27}{t^2} - t = \frac{27 - t^3}{t^2}. \]

Thus, the limit becomes:

\[ \lim_{t \to 3} \frac{t^4 - 12t^2 + 27}{27 - t^3}. \]

### Step 5: Substitute \( t = 3 \)

Now substituting \( t = 3 \):

**Numerator:**
\[ 3^4 - 12 \cdot 3^2 + 27 = 81 - 108 + 27 = 0. \]

**Denominator:**
\[ 27 - 3^3 = 27 - 27 = 0. \]

Again, we have an indeterminate form \( \frac{0}{0} \).

### Step 6: Apply L'Hôpital's Rule

We apply L'Hôpital's Rule, which states that we can take the derivative of the numerator and denominator:

**Numerator's derivative:**
\[ \frac{d}{dt}(t^4 - 12t^2 + 27) = 4t^3 - 24t. \]

**Denominator's derivative:**
\[ \frac{d}{dt}(27 - t^3) = -3t^2. \]

Now we have:

\[ \lim_{t \to 3} \frac{4t^3 - 24t}{-3t^2}. \]

### Step 7: Substitute \( t = 3 \) again

Substituting \( t = 3 \):

**Numerator:**
\[ 4(3^3) - 24(3) = 4(27) - 72 = 108 - 72 = 36. \]

**Denominator:**
\[ -3(3^2) = -3(9) = -27. \]

Thus, the limit becomes:

\[ \frac{36}{-27} = -\frac{4}{3}. \]

### Final Answer

The limit evaluates to:

\[ \lim_{x \to 2} \frac{3^x + 3^{3-x} - 12}{3^{3-x} - 3^{x/2}} = -\frac{4}{3}. \]