Question

question evaluate the limit: $lim_{x
ightarrow25}\frac{50 - 2x}{5-sqrt{x}}$

Explanation:

Step1: Rationalize the denominator

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

$$\begin{align*} \lim_{x ightarrow25}\frac{50 - 2x}{5-\sqrt{x}}\times\frac{5+\sqrt{x}}{5+\sqrt{x}}&=\lim_{x ightarrow25}\frac{(50 - 2x)(5+\sqrt{x})}{25 - x}\\ \end{align*}$$

\]

Step2: Factor out - 2 from the numerator

Factor $50 - 2x=-2(x - 25)$:
\[

$$\begin{align*} \lim_{x ightarrow25}\frac{-2(x - 25)(5+\sqrt{x})}{25 - x}&=\lim_{x ightarrow25}\frac{-2(x - 25)(5+\sqrt{x})}{-(x - 25)}\\ \end{align*}$$

\]

Step3: Cancel out the common factor

Cancel out $(x - 25)$:
\[

$$\begin{align*} \lim_{x ightarrow25}\frac{-2(x - 25)(5+\sqrt{x})}{-(x - 25)}&=\lim_{x ightarrow25}2(5+\sqrt{x}) \end{align*}$$

\]

Step4: Substitute the value of x

Substitute $x = 25$:
\[

$$\begin{align*} 2(5+\sqrt{25})&=2(5 + 5)\\ &=2\times10=10 \end{align*}$$

\]

Answer:

$10$