Question

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

Explanation:

Step1: Rationalize the denominator

Multiply numerator and denominator by $5 + \sqrt{x}$.
\[

$$\begin{align*} \lim_{x ightarrow25}\frac{25 - x}{5-\sqrt{x}}&=\lim_{x ightarrow25}\frac{(25 - x)(5+\sqrt{x})}{(5-\sqrt{x})(5+\sqrt{x})}\\ \end{align*}$$

\]
Since $(a - b)(a + b)=a^{2}-b^{2}$, the denominator is $25 - x$.
So we have $\lim_{x
ightarrow25}\frac{(25 - x)(5+\sqrt{x})}{25 - x}$.

Step2: Simplify the expression

Cancel out the common factor $25 - x$ (for $x
eq25$).
We get $\lim_{x
ightarrow25}(5+\sqrt{x})$.

Step3: Substitute the value of x

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

Answer:

$10$