L'Hôpital's rule does not help with the limit below. Find the limit some other way.
\[
\lim _{x \rightarrow \infty} \frac{\sqrt{25 x+5}}{\sqrt{x+5}}
\]

Step 1 ：The given limit is \(\lim _{x \rightarrow \infty} \frac{\sqrt{25 x+5}}{\sqrt{x+5}}\).

Step 2 ：This limit is in the form of \(\frac{\infty}{\infty}\), which is an indeterminate form. However, L'Hôpital's rule does not apply here because the functions in the numerator and denominator are not differentiable at \(x = \infty\).

Step 3 ：We can simplify the expression by multiplying the numerator and denominator by \(\frac{1}{\sqrt{x}}\). This will give us a simpler expression that we can evaluate the limit of as \(x\) approaches infinity.

Step 4 ：The simplified expression is \(\sqrt{5}\sqrt{5x + 1}\sqrt{\frac{1}{x}}/\sqrt{x + 5}\).

Step 5 ：However, this does not seem correct. There might be a mistake in the simplification process. Let's try to simplify the expression manually and then calculate the limit again.

Step 6 ：The manually simplified expression is \(\sqrt{\frac{25x + 5}{x + 5}}\).

Step 7 ：The limit of the manually simplified expression as \(x\) approaches infinity is 5.

Step 8 ：Final Answer: The limit of the given expression as \(x\) approaches infinity is \(\boxed{5}\).