To solve the given problem, we need to simplify the expression:
[tex]\[
\frac{\frac{x}{25} - \frac{1}{x}}{1 + \frac{5}{x}}
\][/tex]
and then determine which of the given choices it matches.
Let's first rewrite the given expression in a more manageable form:
1. Simplify the numerator [tex]\(\frac{x}{25} - \frac{1}{x}\)[/tex]:
[tex]\[
\frac{x}{25} - \frac{1}{x} = \frac{x^2 - 25}{25x}
\][/tex]
2. Simplify the denominator [tex]\(1 + \frac{5}{x}\)[/tex]:
[tex]\[
1 + \frac{5}{x} = \frac{x + 5}{x}
\][/tex]
Now, the original expression becomes:
[tex]\[
\frac{\frac{x^2 - 25}{25x}}{\frac{x + 5}{x}}
\][/tex]
To divide two fractions, we multiply by the reciprocal of the denominator:
[tex]\[
\frac{x^2 - 25}{25x} \times \frac{x}{x + 5}
\][/tex]
Multiply the fractions:
[tex]\[
\frac{(x^2 - 25)x}{25x (x + 5)} = \frac{x^2 - 25}{25 (x + 5)}
\][/tex]
Notice that the numerator [tex]\(x^2 - 25\)[/tex] can be factored as a difference of squares:
[tex]\[
x^2 - 25 = (x - 5)(x + 5)
\][/tex]
So, we have:
[tex]\[
\frac{(x - 5)(x + 5)}{25 (x + 5)}
\][/tex]
We can cancel out the common term [tex]\((x + 5)\)[/tex]:
[tex]\[
\frac{(x - 5)(\cancel{x + 5})}{25 (\cancel{x + 5})} = \frac{x - 5}{25}
\][/tex]
Thus, the simplified expression is:
[tex]\[
\frac{x - 5}{25}
\][/tex]
Comparing this result with the given choices, we see that it matches:
[tex]\[
\frac{x - 5}{25}
\][/tex]
Therefore, the correct choice is:
[tex]\[
\boxed{3}
\][/tex]
