You've provided a limit problem involving a square root expression and a linear expression, specifically:

$\lim_{x \to 5} \frac{\sqrt{7x+5} - \sqrt{3x+25}}{3(x-5)}$

To solve this limit, we observe that directly substituting $x = 5$ into the expression results in an indeterminate form $\frac{0}{0}$. To resolve this, we can apply L'Hôpital's Rule, which states that for functions $f(x)$ and $g(x)$ that are differentiable near $a$ and for which $\lim_{x \to a} f(x) = \lim_{x \to a} g(x) = 0$ or $\pm \infty$, the limit of $\frac{f(x)}{g(x)}$ as $x$ approaches $a$ can be computed as $\lim_{x \to a} \frac{f'(x)}{g'(x)}$ if this limit exists.

Let's calculate this using differentiation.

I'll first compute the derivatives of the numerator and the denominator, and then re-evaluate the limit.