evaluate the limit, if it exists. (if an answer does not exist, enter dne.)\n lim_{x\rightarrow25}\frac{25…

Explanation:

Step1: Rationalize the denominator

Multiply numerator and denominator by $5 + \sqrt{x}$. [ \begin{align*} \lim_{x\rightarrow25}\frac{25 - x}{5-\sqrt{x}}&=\lim_{x\rightarrow25}\frac{(25 - x)(5+\sqrt{x})}{(5-\sqrt{x})(5+\sqrt{x})}\ \end{align*} ] Since $(a - b)(a + b)=a^{2}-b^{2}$, the denominator is $25 - x$. So we have $\lim_{x\rightarrow25}\frac{(25 - x)(5+\sqrt{x})}{25 - x}$.

Step2: Simplify the expression

Cancel out the common factor $25 - x$ (for $x\neq25$). We get $\lim_{x\rightarrow25}(5+\sqrt{x})$.

Step3: Substitute the value of x

Substitute $x = 25$ into $5+\sqrt{x}$. $5+\sqrt{25}=5 + 5=10$.

Answer:

$10$