Find all possible values of `a` if `lim_(x to a)(x^(9)-a^(9))/(x-a)=9.`

Options

A. `a=pm1`
B. `a=pm2`
C. `a=0`
D. none of the above

Answer

To solve the problem, we need to find all possible values of \( a \) such that:

\[ \lim_{x \to a} \frac{x^9 - a^9}{x - a} = 9. \]

### Step 1: Rewrite the expression using the difference of cubes
We can use the identity for the difference of cubes. The expression \( x^9 - a^9 \) can be factored as follows:

\[ x^9 - a^9 = (x - a)(x^8 + x^7a + x^6a^2 + x^5a^3 + x^4a^4 + x^3a^5 + x^2a^6 + xa^7 + a^8). \]

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

\[ \lim_{x \to a} \frac{(x - a)(x^8 + x^7a + x^6a^2 + x^5a^3 + x^4a^4 + x^3a^5 + x^2a^6 + xa^7 + a^8)}{x - a}. \]

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

\[ \lim_{x \to a} (x^8 + x^7a + x^6a^2 + x^5a^3 + x^4a^4 + x^3a^5 + x^2a^6 + xa^7 + a^8). \]

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

\[ a^8 + a^7a + a^6a^2 + a^5a^3 + a^4a^4 + a^3a^5 + a^2a^6 + aa^7 + a^8 = 9a^8. \]

### Step 5: Set the limit equal to 9
Now we set this equal to 9 (as given in the problem):

\[ 9a^8 = 9. \]

### Step 6: Solve for \( a \)
Dividing both sides by 9 gives:

\[ a^8 = 1. \]

Taking the eighth root of both sides, we find:

\[ a = 1 \quad \text{or} \quad a = -1. \]

### Conclusion
Thus, the possible values of \( a \) are:

\[ \boxed{1 \text{ and } -1}. \]