Evaluate `lim_(xrarra){(x^(12)-a^(12))/(x-a)}.`

Options

A. `12a`
B. `a^(11)`
C. `12a^(11)`
D. none of these

Explanation

To evaluate the limit

\[ \lim_{x \to a} \frac{x^{12} - a^{12}}{x - a}, \]

we can use the factorization of the difference of powers.

### Step 1: Factor the numerator
The expression \(x^{12} - a^{12}\) can be factored using the formula for the difference of squares repeatedly or using the general formula for the difference of powers:

\[ x^{12} - a^{12} = (x - a)(x^{11} + x^{10}a + x^9a^2 + \ldots + x^2a^9 + xa^{10} + a^{11}). \]

### Step 2: Substitute the factorization into the limit
Now, substituting the factorization into the limit gives us:

\[ \lim_{x \to a} \frac{(x - a)(x^{11} + x^{10}a + x^9a^2 + \ldots + x^2a^9 + xa^{10} + a^{11})}{x - a}. \]

### Step 3: Cancel the common terms
Since \(x - a\) is in both the numerator and denominator, we can cancel it (as long as \(x \neq a\)):

\[ \lim_{x \to a} (x^{11} + x^{10}a + x^9a^2 + \ldots + x^2a^9 + xa^{10} + a^{11}). \]

### Step 4: Evaluate the limit
Now we can directly substitute \(x = a\) into the remaining expression:

\[ = a^{11} + a^{10}a + a^9a^2 + \ldots + a^2a^9 + aa^{10} + a^{11}. \]

This simplifies to:

\[ = a^{11} + a^{11} + a^{11} + a^{11} + a^{11} + a^{11} = 12a^{11}. \]

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

\[ \lim_{x \to a} \frac{x^{12} - a^{12}}{x - a} = 12a^{11}. \]