To solve the limit [tex]\(\lim_{x \rightarrow a} \frac{x^9 - a^9}{x - a}\)[/tex], we can follow these steps:
1. Recognize the form: The expression [tex]\(\frac{x^9 - a^9}{x - a}\)[/tex] resembles the derivative of a polynomial at a certain point. Specifically, it looks like the definition of the derivative of [tex]\(f(x) = x^9\)[/tex] at [tex]\(x = a\)[/tex].
2. Factor the numerator: To make it easier to work with the expression, we factor [tex]\(x^9 - a^9\)[/tex]. Using the difference of powers formula, we have:
[tex]\[
x^9 - a^9 = (x - a)(x^8 + x^7a + x^6a^2 + \cdots + x a^7 + a^8)
\][/tex]
3. Simplify the limit: Substitute this factorization into the limit expression:
[tex]\[
\frac{x^9 - a^9}{x - a} = \frac{(x - a)(x^8 + x^7a + x^6a^2 + \cdots + x a^7 + a^8)}{x - a}
\][/tex]
Since [tex]\(x \neq a\)[/tex] simplifies the fraction to:
[tex]\[
x^8 + x^7a + x^6a^2 + \cdots + x a^7 + a^8
\][/tex]
4. Evaluate the limit: Now, we need to take the limit of this polynomial as [tex]\(x\)[/tex] approaches [tex]\(a\)[/tex]:
[tex]\[
\lim_{x \rightarrow a} (x^8 + x^7a + x^6a^2 + \cdots + x a^7 + a^8)
\][/tex]
When [tex]\(x\)[/tex] approaches [tex]\(a\)[/tex], each term involving [tex]\(x\)[/tex] will simply become a form including [tex]\(a\)[/tex]:
[tex]\[
a^8 + a^7a + a^6a^2 + a^5a^3 + a^4a^4 + a^3a^5 + a^2a^6 + a a^7 + a^8
\][/tex]
Which simplifies to:
[tex]\[
a^8 + a^8 + a^8 + a^8 + a^8 + a^8 + a^8 + a^8 + a^8 = 9a^8
\][/tex]
Therefore, the limit is:
[tex]\[
\lim_{x \rightarrow a} \frac{x^9 - a^9}{x - a} = 9a^8
\][/tex]
5. Given condition: The problem states that this limit equals 9. Thus, we equate our result:
[tex]\[
9a^8 = 9
\][/tex]
Solving for [tex]\(a\)[/tex], we divide both sides by 9:
[tex]\[
a^8 = 1
\][/tex]
Taking the eighth root of both sides:
[tex]\[
a = 1
\][/tex]
Thus, the detailed, step-by-step solution shows that the limit [tex]\(\lim_{x \rightarrow a} \frac{x^9 - a^9}{x - a}\)[/tex] simplifies to [tex]\(9a^8\)[/tex], and given that this limit equals 9, we find [tex]\(a = 1\)[/tex].
