Evaluate the following limits :
`Lim_(x to 0) (3^(x)-1)/x `

Answer

To evaluate the limit

\[ \lim_{x \to 0} \frac{3^x - 1}{x}, \]

we can follow these steps:

### Step 1: Identify the form of the limit
As \( x \) approaches 0, the numerator \( 3^x - 1 \) approaches \( 3^0 - 1 = 1 - 1 = 0 \), and the denominator \( x \) approaches 0. Thus, we have the indeterminate form \( \frac{0}{0} \).

**Hint:** Check the values of the numerator and denominator as \( x \) approaches the limit to determine if it's an indeterminate form.

### Step 2: Apply L'Hôpital's Rule
Since we have the \( \frac{0}{0} \) form, we can apply L'Hôpital's Rule, which states that if the limit results in an indeterminate form, we can differentiate the numerator and the denominator separately.

**Hint:** Remember that L'Hôpital's Rule can be applied when you encounter \( \frac{0}{0} \) or \( \frac{\infty}{\infty} \) forms.

### Step 3: Differentiate the numerator and denominator
- Differentiate the numerator \( 3^x - 1 \):
\[ \frac{d}{dx}(3^x - 1) = 3^x \ln(3). \]
- Differentiate the denominator \( x \):
\[ \frac{d}{dx}(x) = 1. \]

**Hint:** Use the formula for the derivative of \( a^x \) which is \( a^x \ln(a) \).

### Step 4: Rewrite the limit using the derivatives
Now we can rewrite the limit using the derivatives:
\[ \lim_{x \to 0} \frac{3^x \ln(3)}{1}. \]

**Hint:** After applying L'Hôpital's Rule, substitute the limit directly into the new expression.

### Step 5: Evaluate the limit
Substituting \( x = 0 \) into the expression gives:
\[ 3^0 \ln(3) = 1 \cdot \ln(3) = \ln(3). \]

**Hint:** Remember that \( 3^0 = 1 \) when evaluating the limit.

### Final Answer
Thus, the limit is:
\[ \lim_{x \to 0} \frac{3^x - 1}{x} = \ln(3). \]