Let's analyze the limit problem step-by-step to find all possible values of [tex]\( a \)[/tex].
Given the limit expression:
[tex]\[
\lim_{x \to a} \frac{x^9 - a^9}{x - a} = 9
\][/tex]
we can start by recognizing that the numerator [tex]\( x^9 - a^9 \)[/tex] can be factored using the difference of powers formula. Specifically:
[tex]\[
x^9 - a^9 = (x - a)(x^8 + x^7 a + x^6 a^2 + x^5 a^3 + x^4 a^4 + x^3 a^5 + x^2 a^6 + x a^7 + a^8)
\][/tex]
So, we can rewrite the limit expression as:
[tex]\[
\lim_{x \to a} \frac{(x - a)(x^8 + x^7 a + x^6 a^2 + x^5 a^3 + x^4 a^4 + x^3 a^5 + x^2 a^6 + x a^7 + a^8)}{x - a}
\][/tex]
The [tex]\( (x - a) \)[/tex] terms cancel out, leaving us with:
[tex]\[
\lim_{x \to a} (x^8 + x^7 a + x^6 a^2 + x^5 a^3 + x^4 a^4 + x^3 a^5 + x^2 a^6 + x a^7 + a^8)
\][/tex]
As [tex]\( x \to a \)[/tex], the expression inside the limit simplifies by substituting [tex]\( x \)[/tex] with [tex]\( a \)[/tex]:
[tex]\[
a^8 + a^7 a + a^6 a^2 + a^5 a^3 + a^4 a^4 + a^3 a^5 + a^2 a^6 + a a^7 + a^8
\][/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]
We are given that this whole expression is equal to 9, thus:
[tex]\[
9a^8 = 9
\][/tex]
Solving for [tex]\( a^8 \)[/tex]:
[tex]\[
a^8 = 1
\][/tex]
Now, we find the 8th roots of unity of 1. These are the solutions to the equation [tex]\( a^8 = 1 \)[/tex]. The 8th roots of unity are:
[tex]\[
a = e^{2\pi i k / 8} \quad \text{for } k = 0, 1, 2, 3, 4, 5, 6, 7
\][/tex]
Calculating these, we have:
- [tex]\( k = 0 \)[/tex]: [tex]\( a = e^{0 \cdot \pi i / 4} = 1 \)[/tex]
- [tex]\( k = 1 \)[/tex]: [tex]\( a = e^{\pi i / 4} = \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \)[/tex]
- [tex]\( k = 2 \)[/tex]: [tex]\( a = e^{\pi i / 2} = i \)[/tex]
- [tex]\( k = 3 \)[/tex]: [tex]\( a = e^{3\pi i / 4} = -\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2} \)[/tex]
- [tex]\( k = 4 \)[/tex]: [tex]\( a = e^{\pi i} = -1 \)[/tex]
- [tex]\( k = 5 \)[/tex]: [tex]\( a = e^{5\pi i / 4} = -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} \)[/tex]
- [tex]\( k = 6 \)[/tex]: [tex]\( a = e^{3\pi i / 2} = -i \)[/tex]
- [tex]\( k = 7 \)[/tex]: [tex]\( a = e^{7\pi i / 4} = \frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2} \)[/tex]
So, the possible values of [tex]\( a \)[/tex] are:
[tex]\[
\boxed{1, -1, i, -i, \frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}, -\frac{\sqrt{2}}{2} - i\frac{\sqrt{2}}{2}}
\][/tex]
