`lim_(xto0) ((1-3^(x)-4^(x)+12^(x)))/(sqrt((2cosx+7))-3)`

Explanation

To solve the limit problem

\[ \lim_{x \to 0} \frac{1 - 3^x - 4^x + 12^x}{\sqrt{2 \cos x + 7} - 3}, \]

we will follow these steps:

### Step 1: Evaluate the limit directly
First, we substitute \(x = 0\) into the expression:

\[ 1 - 3^0 - 4^0 + 12^0 = 1 - 1 - 1 + 1 = 0, \]
and for the denominator:

\[ \sqrt{2 \cos(0) + 7} - 3 = \sqrt{2 \cdot 1 + 7} - 3 = \sqrt{9} - 3 = 3 - 3 = 0. \]

Since both the numerator and denominator approach 0, we have a \( \frac{0}{0} \) indeterminate form.

### Step 2: Simplify the expression
We can simplify the numerator. Notice that \(12^x = (3 \cdot 4)^x = 3^x \cdot 4^x\). Thus, we can rewrite the numerator:

\[ 1 - 3^x - 4^x + 12^x = 1 - 3^x - 4^x + 3^x \cdot 4^x. \]

### Step 3: Factor the numerator
We can factor the numerator as follows:

\[ 1 - 3^x - 4^x + 3^x \cdot 4^x = (1 - 3^x)(1 - 4^x). \]

### Step 4: Rationalize the denominator
Next, we rationalize the denominator:

\[ \sqrt{2 \cos x + 7} - 3 = \frac{(2 \cos x + 7) - 9}{\sqrt{2 \cos x + 7} + 3} = \frac{2 \cos x - 2}{\sqrt{2 \cos x + 7} + 3} = \frac{2(\cos x - 1)}{\sqrt{2 \cos x + 7} + 3}. \]

### Step 5: Substitute back into the limit
Now, substituting back into the limit:

\[ \lim_{x \to 0} \frac{(1 - 3^x)(1 - 4^x)}{\frac{2(\cos x - 1)}{\sqrt{2 \cos x + 7} + 3}} = \lim_{x \to 0} \frac{(1 - 3^x)(1 - 4^x)(\sqrt{2 \cos x + 7} + 3)}{2(\cos x - 1)}. \]

### Step 6: Apply L'Hôpital's Rule or use known limits
We can apply the known limits:

\[ \lim_{x \to 0} \frac{1 - a^x}{x} = -\ln a, \]

for \(a = 3\) and \(a = 4\):

\[ \lim_{x \to 0} \frac{1 - 3^x}{x} = -\ln 3, \]
\[ \lim_{x \to 0} \frac{1 - 4^x}{x} = -\ln 4. \]

### Step 7: Evaluate the limit
Now we can evaluate the limit:

\[ \lim_{x \to 0} \frac{(-\ln 3)(-\ln 4)(\sqrt{2 \cdot 1 + 7} + 3)}{2 \cdot (-1)} = \frac{\ln 3 \cdot \ln 4 \cdot (3 + 3)}{2} = \frac{6 \ln 3 \cdot \ln 4}{2} = 3 \ln 3 \cdot \ln 4. \]

### Final Answer
Thus, the final answer is:

\[ \lim_{x \to 0} \frac{1 - 3^x - 4^x + 12^x}{\sqrt{2 \cos x + 7} - 3} = 3 \ln 3 \cdot \ln 4. \]

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