Questions: lim as x approaches 25 of (sqrt(x) - 5) / (2x - 50)

Evaluate the limit:
\[ \lim _{x \rightarrow 25} \frac{\sqrt{x}-5}{2x-50} \]

Simplify the denominator

Factor the denominator:
\[ 2x - 50 = 2(x - 25) \]

Rationalize the numerator

Multiply the numerator and denominator by the conjugate of the numerator:
\[ \frac{\sqrt{x}-5}{2(x-25)} \cdot \frac{\sqrt{x}+5}{\sqrt{x}+5} = \frac{(\sqrt{x}-5)(\sqrt{x}+5)}{2(x-25)(\sqrt{x}+5)} \]

Simplify the numerator

Use the difference of squares formula:
\[ (\sqrt{x}-5)(\sqrt{x}+5) = x - 25 \]

Cancel common terms

Substitute the simplified numerator and cancel \(x - 25\):
\[ \frac{x - 25}{2(x-25)(\sqrt{x}+5)} = \frac{1}{2(\sqrt{x}+5)} \]

Evaluate the limit

Substitute \(x = 25\) into the simplified expression:
\[ \frac{1}{2(\sqrt{25}+5)} = \frac{1}{2(5+5)} = \frac{1}{20} \]

The limit evaluates to:
\[ \boxed{\frac{1}{20}} \]

The final answer is:
\[ \boxed{\frac{1}{20}} \]