In sub - part (i) to (x) choose the correct option and in sub - part (xi) to (xy), answer the questions as intructed .
If `lim_(xtoa)(x^(9)-a^(9))/(x-a)=lim_(xto5)(4+x)`, then a equals :

Options

A. `+-1`
B. `-1`
C. 1
D. 0

Answer

To solve the problem, we need to evaluate the limit and find the value of \( a \).

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

### Step 1: Simplify the left-hand side using the limit definition
The expression \( \frac{x^9 - a^9}{x - a} \) can be simplified using the formula for the difference of powers:
\[ x^9 - a^9 = (x - a)(x^8 + x^7 a + x^6 a^2 + x^5 a^3 + x^4 a^4 + x^3 a^5 + x^2 a^6 + x a^7 + a^8) \]
Thus,
\[ \frac{x^9 - a^9}{x - a} = x^8 + x^7 a + x^6 a^2 + x^5 a^3 + x^4 a^4 + x^3 a^5 + x^2 a^6 + x a^7 + a^8 \]

### Step 2: Evaluate the limit as \( x \to a \)
Taking the limit as \( x \to a \):
\[ \lim_{x \to a} \frac{x^9 - a^9}{x - a} = a^8 + a^7 a + a^6 a^2 + a^5 a^3 + a^4 a^4 + a^3 a^5 + a^2 a^6 + a a^7 + a^8 = 9a^8 \]

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

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

### Step 5: Solve for \( a \)
Dividing both sides by 9 gives:
\[ a^8 = 1 \]
Taking the eighth root of both sides, we find:
\[ a = 1 \]

### Final Answer
Thus, the value of \( a \) is:
\[ \boxed{1} \]