Compute `lim_(x rarr 0)(e^(3x)-1)/x`

Explanation

To compute the limit \(\lim_{x \to 0} \frac{e^{3x} - 1}{x}\), we can follow these steps:

### Step 1: Recognize the form of the limit
As \(x\) approaches 0, both the numerator \(e^{3x} - 1\) and the denominator \(x\) approach 0. This gives us an indeterminate form of \(\frac{0}{0}\).

### Step 2: Use the known limit identity
We know from calculus that:
\[ \lim_{u \to 0} \frac{e^u - 1}{u} = 1 \]
We can manipulate our limit to use this identity.

### Step 3: Rewrite the limit
We can express our limit in terms of the known limit identity by substituting \(u = 3x\). Therefore, as \(x \to 0\), \(u \to 0\) as well.

We rewrite the limit:
\[ \lim_{x \to 0} \frac{e^{3x} - 1}{x} = \lim_{x \to 0} \frac{e^{3x} - 1}{3x} \cdot 3 \]

### Step 4: Apply the limit identity
Now we can apply the limit identity:
\[ \lim_{x \to 0} \frac{e^{3x} - 1}{3x} = 1 \]
Thus, we have:
\[ \lim_{x \to 0} \frac{e^{3x} - 1}{x} = 3 \cdot 1 = 3 \]

### Conclusion
Therefore, the limit is:
\[ \lim_{x \to 0} \frac{e^{3x} - 1}{x} = 3 \]

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