Question

evaluate the limit, if it exists. (if an answer does not exist, enter dne.)
\\(lim_{x
ightarrow25}\frac{5 - sqrt{x}}{25x - x^{2}}\\)

Explanation:

Step1: Factor the denominator

Factor \(25x - x^{2}\) as \(x(25 - x)\). So the limit becomes \(\lim_{x
ightarrow25}\frac{5-\sqrt{x}}{x(25 - x)}\).

Step2: Rationalize the numerator

Multiply the numerator and denominator by the conjugate of the numerator \(5+\sqrt{x}\). We get \(\lim_{x
ightarrow25}\frac{(5 - \sqrt{x})(5+\sqrt{x})}{x(25 - x)(5+\sqrt{x})}\).
Using the difference - of - squares formula \((a - b)(a + b)=a^{2}-b^{2}\), the numerator is \(25 - x\). So the limit is \(\lim_{x
ightarrow25}\frac{25 - x}{x(25 - x)(5+\sqrt{x})}\).

Step3: Simplify the expression

Cancel out the common factor \((25 - x)\) (since \(x
eq25\) when taking the limit), we have \(\lim_{x
ightarrow25}\frac{1}{x(5+\sqrt{x})}\).

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:

\(\frac{1}{250}\)