Find all possible values of 'a', if :
`lim_(x to a)(x^(9)-a^(9))/(x-a)=lim_(x to 5)(4+x)`

Answer

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

\[ \lim_{x \to a} \frac{x^9 - a^9}{x - a} = \lim_{x \to 5} (4 + x) \]

### Step 1: Evaluate the right-hand side limit
First, we evaluate the limit on the right-hand side:

\[ \lim_{x \to 5} (4 + x) = 4 + 5 = 9 \]

### Step 2: Rewrite the left-hand side limit
Now we rewrite the left-hand side limit:

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

Using the formula for the difference of powers, we know that:

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

Thus, we can rewrite the limit as:

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

### Step 3: Simplify the limit
Since \( x - a \) cancels out, we have:

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

Now we can substitute \( x = a \):

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

This simplifies to:

\[ = 9a^8 \]

### Step 4: Set the limits equal to each other
Now we set the two limits equal to each other:

\[ 9a^8 = 9 \]

### Step 5: Solve for \( a \)
Dividing both sides by 9 gives:

\[ a^8 = 1 \]

Taking the eighth root of both sides, we find:

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

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

\[ a = 1 \quad \text{and} \quad a = -1 \]