If `Lt_(x to a)(x^(9)-a^(9))/(x-a)=Lt_(xto5)(x+4)` then all possible values of a are
(i) 2, 3
(ii) -2, 2
(iii) -1, 1
(iv) -3, 3

Options

A. 2,3
B. `-2,2`
C. `-1,1`
D. `-3,3`

Answer

To solve the problem, we need to evaluate the limit expression given in the question. Let's break it down step by step.

### Step 1: Write the limit expression
We have:
\[ \lim_{x \to a} \frac{x^9 - a^9}{x - a} = \lim_{x \to 5} (x + 4) \]

### Step 2: Apply L'Hôpital's Rule
Since both the numerator and denominator approach 0 as \( x \) approaches \( a \), we can apply L'Hôpital's Rule. This states that if we have a limit of the form \( \frac{0}{0} \), we can take the derivative of the numerator and the denominator.

Taking the derivative:
- Derivative of the numerator \( x^9 - a^9 \) with respect to \( x \) is \( 9x^8 \).
- Derivative of the denominator \( x - a \) with respect to \( x \) is \( 1 \).

Thus, we can rewrite the limit as:
\[ \lim_{x \to a} \frac{9x^8}{1} = 9a^8 \]

### Step 3: Evaluate the right-hand side limit
Now, we evaluate the right-hand side:
\[ \lim_{x \to 5} (x + 4) = 5 + 4 = 9 \]

### Step 4: Set the two limits equal to each other
Now we can set the two limits equal to each other:
\[ 9a^8 = 9 \]

### Step 5: Simplify the equation
Dividing both sides by 9 gives:
\[ a^8 = 1 \]

### Step 6: Solve for \( a \)
The equation \( a^8 = 1 \) implies:
\[ a = 1 \quad \text{or} \quad a = -1 \]

### Conclusion
Thus, the possible values of \( a \) are \( 1 \) and \( -1 \).

The final answer is:
\[ \text{All possible values of } a \text{ are } -1 \text{ and } 1. \]

### Answer Options
The correct option is (iii) -1, 1.

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