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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.

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

Solve the limit lim_{x \to 5} (\sqrt{7x+5} - \sqrt{3x+25}) / (3(x-5)) using L'Hôpital's Rule. This problem explores the concept of indeterminate forms and differentiation of square root functions. A detailed step-by-step solution is provided. Suitable for college-level calculus students.

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