Question
If limit x-->a x^9-a^9/x-a=9 find a
Answers
GPT-4o mini
To solve the limit problem
\[
\lim_{x \to a} \frac{x^9 - a^9}{x - a} = 9,
\]
we can use the fact that the expression \(\frac{x^9 - a^9}{x - a}\) can be expressed using the derivative. Specifically, the limit can be evaluated as follows:
By applying L'Hôpital's Rule, which is applicable for a \(0/0\) form, we take the derivative of the numerator and the denominator.
1. Derivative of the numerator (\(x^9 - a^9\)):
\[
\frac{d}{dx}(x^9 - a^9) = 9x^8.
\]
2. Derivative of the denominator (\(x - a\)):
\[
\frac{d}{dx}(x - a) = 1.
\]
So, applying L'Hôpital's Rule, we have:
\[
\lim_{x \to a} \frac{x^9 - a^9}{x - a} = \lim_{x \to a} \frac{9x^8}{1} = 9a^8.
\]
Now we can set the expression equal to the limit value we are given:
\[
9a^8 = 9.
\]
Dividing both sides by 9 gives:
\[
a^8 = 1.
\]
Taking the eighth root of both sides, we find:
\[
a = 1 \quad \text{or} \quad a = -1.
\]
Thus, the possible values of \(a\) are:
\[
\boxed{1 \text{ and } -1}.
\]
\[
\lim_{x \to a} \frac{x^9 - a^9}{x - a} = 9,
\]
we can use the fact that the expression \(\frac{x^9 - a^9}{x - a}\) can be expressed using the derivative. Specifically, the limit can be evaluated as follows:
By applying L'Hôpital's Rule, which is applicable for a \(0/0\) form, we take the derivative of the numerator and the denominator.
1. Derivative of the numerator (\(x^9 - a^9\)):
\[
\frac{d}{dx}(x^9 - a^9) = 9x^8.
\]
2. Derivative of the denominator (\(x - a\)):
\[
\frac{d}{dx}(x - a) = 1.
\]
So, applying L'Hôpital's Rule, we have:
\[
\lim_{x \to a} \frac{x^9 - a^9}{x - a} = \lim_{x \to a} \frac{9x^8}{1} = 9a^8.
\]
Now we can set the expression equal to the limit value we are given:
\[
9a^8 = 9.
\]
Dividing both sides by 9 gives:
\[
a^8 = 1.
\]
Taking the eighth root of both sides, we find:
\[
a = 1 \quad \text{or} \quad a = -1.
\]
Thus, the possible values of \(a\) are:
\[
\boxed{1 \text{ and } -1}.
\]