Bbluncnutb npenen lim _(xarrow 10)(-sqrt (10)+sqrt (x))/(x-10)

To evaluate the limit , we can use algebraic manipulation and L'Hôpital's Rule.First, let's rewrite the expression: As approaches 10, both the numerator and the denominator approach 0, which means we can apply L'Hôpital's Rule. L'Hôpital's Rule states that if \(\lim_{x \to c} \frac{f(x)}{g(x)}\) results in an indeterminate form , then: provided the limit on the right-hand side exists.Let \(f(x) = -\sqrt{10} + \sqrt{x}\) and \(g(x) = x - 10\).First, we find the derivatives of \(f(x)\) and \(g(x)\): Now, we apply L'Hôpital's Rule: As approaches 10: Therefore, the limit is: This can also be written as: So, the final\lim_{x \frac{-\sqrt{10} + \sqrt{x}}{x - 10} = \frac{\sqrt{10}}{20}\]