evaluate the limit, if it exists. (if an answer does not exist, enter dne.)\n\\(lim_{x\rightarrow25}\frac{5…

Explanation:

Step1: Factor the denominator

Factor (25x - x^{2}) as (x(25 - x)). So the limit becomes (\lim_{x\rightarrow25}\frac{5-\sqrt{x}}{x(25 - x)}).

Step2: Rationalize the numerator

Multiply the numerator and denominator by the conjugate of the numerator (5+\sqrt{x}). We get (\lim_{x\rightarrow25}\frac{(5 - \sqrt{x})(5+\sqrt{x})}{x(25 - x)(5+\sqrt{x})}). Using the difference - of - squares formula ((a - b)(a + b)=a^{2}-b^{2}), the numerator is (25 - x). So the limit is (\lim_{x\rightarrow25}\frac{25 - x}{x(25 - x)(5+\sqrt{x})}).

Step3: Simplify the expression

Cancel out the common factor ((25 - x)) (since (x\neq25) when taking the limit), we have (\lim_{x\rightarrow25}\frac{1}{x(5+\sqrt{x})}).

Step4: Substitute (x = 25)

Substitute (x = 25) into (\frac{1}{x(5+\sqrt{x})}), we get (\frac{1}{25(5 + \sqrt{25})}=\frac{1}{25\times(5 + 5)}=\frac{1}{250}).

Answer:

(\frac{1}{250})