To find the limit [tex]\(\lim _{x \rightarrow 8} \frac{x^9}{8 x}\)[/tex], let's go through the problem step by step:
1. Simplify the Expression:
The given function is:
[tex]\[
\frac{x^9}{8x}
\][/tex]
We can simplify this expression by dividing [tex]\(x^9\)[/tex] by [tex]\(8x\)[/tex]:
[tex]\[
\frac{x^9}{8x} = \frac{x^{9}}{8} \times \frac{1}{x} = \frac{x^{8}}{8}
\][/tex]
So the simplified form of the expression is:
[tex]\[
\frac{x^8}{8}
\][/tex]
2. Evaluate the Limit:
Now, we need to evaluate the limit of the simplified expression as [tex]\(x\)[/tex] approaches 8:
[tex]\[
\lim_{x \to 8} \frac{x^8}{8}
\][/tex]
Substitute [tex]\(x = 8\)[/tex] into the simplified expression:
[tex]\[
\frac{8^8}{8}
\][/tex]
3. Compute the Value:
Calculate [tex]\(8^8\)[/tex]:
[tex]\[
8^8 = 16777216
\][/tex]
Now, divide by 8:
[tex]\[
\frac{16777216}{8} = 2097152
\][/tex]
Therefore, the limit is:
[tex]\[
\boxed{2097152}
\][/tex]
