If `lim_(x to 0) (x^(9) + a^(9))/(x + a) = 9`, find the value of a

Answer

To solve the limit problem given by

\[ \lim_{x \to 0} \frac{x^9 + a^9}{x + a} = 9, \]

we will follow these steps:

### Step 1: Substitute \( x = 0 \) in the limit expression

When we substitute \( x = 0 \) into the limit, we get:

\[ \frac{0^9 + a^9}{0 + a} = \frac{a^9}{a} = a^8. \]

### Step 2: Set the limit equal to 9

From the limit condition, we have:

\[ a^8 = 9. \]

### Step 3: Solve for \( a \)

To find \( a \), we take the eighth root of both sides:

\[ a = 9^{1/8}. \]

### Step 4: Consider both positive and negative roots

Since \( 9 = 3^2 \), we can express \( 9^{1/8} \) as:

\[ a = (3^2)^{1/8} = 3^{1/4}. \]

Thus, the possible values for \( a \) are:

\[ a = 3^{1/4} \quad \text{or} \quad a = -3^{1/4}. \]

### Final Answer

The values of \( a \) are:

\[ a = \pm 3^{1/4}. \]

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