Evaluate `lim_(x to 1) (x^(3) - 1)/(x - 1)`

Explanation

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

### Step 1: Identify the limit
We start with the limit expression:
\[ L = \lim_{x \to 1} \frac{x^3 - 1}{x - 1} \]

### Step 2: Factor the numerator
Notice that \(x^3 - 1\) can be factored using the difference of cubes formula:
\[ a^3 - b^3 = (a - b)(a^2 + ab + b^2) \]
Here, \(a = x\) and \(b = 1\), so we can write:
\[ x^3 - 1 = (x - 1)(x^2 + x \cdot 1 + 1^2) = (x - 1)(x^2 + x + 1) \]

### Step 3: Substitute the factorization into the limit
Now, substitute the factorization back into the limit:
\[ L = \lim_{x \to 1} \frac{(x - 1)(x^2 + x + 1)}{x - 1} \]

### Step 4: Cancel the common terms
Since \(x - 1\) appears in both the numerator and the denominator, we can cancel it (as long as \(x \neq 1\)):
\[ L = \lim_{x \to 1} (x^2 + x + 1) \]

### Step 5: Evaluate the limit
Now, we can directly substitute \(x = 1\) into the remaining expression:
\[ L = 1^2 + 1 + 1 = 1 + 1 + 1 = 3 \]

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

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