Evaluate the following limits :
`lim_(x to 0) (3^(2x)-2^(3x))/x`

Answer

To evaluate the limit

\[ \lim_{x \to 0} \frac{3^{2x} - 2^{3x}}{x}, \]

we can follow these steps:

### Step 1: Rewrite the expression
We start by adding and subtracting 1 in the numerator:

\[ \lim_{x \to 0} \frac{3^{2x} - 1 - (2^{3x} - 1)}{x}. \]

### Step 2: Separate the limit
We can separate the limit into two parts:

\[ \lim_{x \to 0} \left( \frac{3^{2x} - 1}{x} - \frac{2^{3x} - 1}{x} \right). \]

### Step 3: Multiply and divide by appropriate factors
Next, we will manipulate each term. For the first term, we multiply and divide by \(2\) and for the second term, we multiply and divide by \(3\):

\[ = \lim_{x \to 0} \left( \frac{3^{2x} - 1}{2x} \cdot 2 - \frac{2^{3x} - 1}{3x} \cdot 3 \right). \]

### Step 4: Apply the limit property
Using the property

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

we can evaluate the limits:

1. For the first term:

\[ \lim_{x \to 0} \frac{3^{2x} - 1}{2x} = \frac{1}{2} \log 3^2 = \log 3. \]

2. For the second term:

\[ \lim_{x \to 0} \frac{2^{3x} - 1}{3x} = \frac{1}{3} \log 2^3 = \log 2. \]

### Step 5: Combine the results
Now we can combine the results:

\[ = 2 \log 3 - 3 \log 2. \]

### Step 6: Use logarithm properties
Using the properties of logarithms:

\[ = \log 3^2 - \log 2^3 = \log \frac{9}{8}. \]

### Final Answer
Thus, the limit evaluates to:

\[ \lim_{x \to 0} \frac{3^{2x} - 2^{3x}}{x} = \log \frac{9}{8}. \]

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