`lim_(x to 0)(3^(2x)-2^(3x))/(x)` is equal to

Options

A. A. 2 log `(3)/(2)`
B. B. 3 log `2/3`
C. C. log `9/8`
D. D. log `8/9`

Answer

To solve the limit \( \lim_{x \to 0} \frac{3^{2x} - 2^{3x}}{x} \), we can follow these steps:

### Step 1: Rewrite the limit
We start with the limit expression:
\[ \lim_{x \to 0} \frac{3^{2x} - 2^{3x}}{x} \]
To simplify, we can add and subtract 1 in the numerator:
\[ \lim_{x \to 0} \frac{(3^{2x} - 1) - (2^{3x} - 1)}{x} \]

### Step 2: Split the limit
We can split the limit into two separate limits:
\[ \lim_{x \to 0} \frac{3^{2x} - 1}{x} - \lim_{x \to 0} \frac{2^{3x} - 1}{x} \]

### Step 3: Use the limit identity
We can use the limit identity \( \lim_{x \to 0} \frac{a^x - 1}{x} = \ln a \) for both terms. For the first term:
\[ \lim_{x \to 0} \frac{3^{2x} - 1}{x} = \lim_{x \to 0} \frac{(3^2)^x - 1}{x} = 2 \ln 3 \]
For the second term:
\[ \lim_{x \to 0} \frac{2^{3x} - 1}{x} = \lim_{x \to 0} \frac{(2^3)^x - 1}{x} = 3 \ln 2 \]

### Step 4: Combine the results
Now we can combine the results from both limits:
\[ 2 \ln 3 - 3 \ln 2 \]

### Step 5: Simplify the expression
We can express this as:
\[ 2 \ln 3 - 3 \ln 2 = \ln(3^2) - \ln(2^3) = \ln\left(\frac{3^2}{2^3}\right) = \ln\left(\frac{9}{8}\right) \]

### Final Answer
Thus, the limit evaluates to:
\[ \lim_{x \to 0} \frac{3^{2x} - 2^{3x}}{x} = \ln\left(\frac{9}{8}\right) \]

### Options
The correct answer corresponds to option C: \( \ln\left(\frac{9}{8}\right) \).

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