To evaluate the limit
[tex]\[
\lim _{x \rightarrow 6} \frac{x-6}{x^2-6 x},
\][/tex]
we begin by analyzing the given expression. First, let's rewrite the denominator, [tex]\(x^2 - 6x\)[/tex].
Notice that [tex]\(x^2 - 6x\)[/tex] can be factored as follows:
[tex]\[
x^2 - 6x = x(x - 6).
\][/tex]
Therefore, the original limit expression can be rewritten as:
[tex]\[
\frac{x-6}{x^2-6 x} = \frac{x-6}{x(x - 6)}.
\][/tex]
At this point, we see that both the numerator and the denominator have a common factor, [tex]\(x - 6\)[/tex]. We can cancel this common factor, as long as [tex]\(x \neq 6\)[/tex]. Cancelling the common factor [tex]\(x - 6\)[/tex] gives us:
[tex]\[
\frac{x-6}{x(x - 6)} = \frac{1}{x}.
\][/tex]
Now, we need to evaluate the limit of the simplified expression as [tex]\(x\)[/tex] approaches 6:
[tex]\[
\lim _{x \rightarrow 6} \frac{1}{x}.
\][/tex]
Since we are now dealing with the continuous function [tex]\(\frac{1}{x}\)[/tex], we can directly substitute [tex]\(x = 6\)[/tex] into the expression:
[tex]\[
\frac{1}{6}.
\][/tex]
Hence, the limit is:
[tex]\[
\lim _{x \rightarrow 6} \frac{x-6}{x^2-6 x} = \frac{1}{6}.
\][/tex]
So, the final answer is:
[tex]\[
\boxed{\frac{1}{6}}
\][/tex]
